{ "meta": { "generated_at": "2026-08-25T17:18:55Z", "counts": { "total": 850, "registry": 30, "overlay": 820, "with_informal": 558, "signed_off": 2, "mathlib_conflicts": 0 }, "mathlib_names": { "source": "https://leanprover-community.github.io/mathlib4_docs/declarations/declaration-data.bmp", "fetched_at": "2026-08-25T15:12:10Z", "declarations": 421326, "sha256": "0ec8f2b2375ddc43df49f183baf70c90b8cf6e3194450777fe9e5a9d14ce7029", "file": "data/mathlib-names.tsv.gz", "note": "Snapshot of current Mathlib docs; entries pin older Mathlib revisions." } }, "records": [ { "id": "artificial-theorems:QLS.Stoch.robbinsSiegmund_expBound", "name": "QLS.Stoch.robbinsSiegmund_expBound", "kind": "theorem", "origin": "registry", "entry": "artificial-theorems", "entry_name": "ArtificialTheorems", "repository": "https://github.com/GasStationManager/ArtificialTheorems", "commit": "9c0f970db29553e3c32bccb25fd13c3f54dc3a8e", "lean": "leanprover/lean4:v4.27.0", "spec_file": "specs/artificial-theorems/Registry/ArtificialTheorems/Opt/RobbinsSiegmund.lean", "impl_module": "ArtificialTheorems.Opt.RobbinsSiegmund", "lines": [ 33, 58 ], "statement": "theorem robbinsSiegmund_expBound.{v}\n {Ω : Type v} [m0 : MeasurableSpace Ω]\n (μ : Measure Ω) [IsFiniteMeasure μ]\n (ℱ : Filtration ℕ m0)\n (X Y Z W : ℕ → Ω → ℝ)\n (adapted_X : Adapted ℱ X) (adapted_Y : Adapted ℱ Y)\n (adapted_Z : Adapted ℱ Z) (adapted_W : Adapted ℱ W)\n (predictable_Y : Adapted ℱ fun t => Y (t + 1))\n (predictable_Z : Adapted ℱ fun t => Z (t + 1))\n (predictable_W : Adapted ℱ fun t => W (t + 1))\n (hX_nonneg : ∀ t ω, 0 ≤ X t ω)\n (hY_nonneg : ∀ t ω, 0 ≤ Y t ω)\n (hZ_nonneg : ∀ t ω, 0 ≤ Z t ω)\n (hW_nonneg : ∀ t ω, 0 ≤ W t ω)\n (condexp_ineq : ∀ t,\n μ[fun ω => X (t + 1) ω | ℱ t]\n ≤ᵐ[μ] fun ω => (1 + Y (t + 1) ω) * X t ω + Z (t + 1) ω - W (t + 1) ω)\n (integrable_X : ∀ t, Integrable (X t) μ)\n (integrable_Z : ∀ t, Integrable (Z t) μ)\n (integrable_W : ∀ t, Integrable (W t) μ)\n (sumEZ : Summable (fun t => ∫ ω, Z t ω ∂μ))\n (prod_bound : ∃ C : ℝ, 0 < C ∧ ∀ t ω, prodY Y t ω ≤ C)\n : ∃ Xlim : Ω → ℝ,\n (∀ᵐ ω ∂μ, Tendsto (fun t => X t ω) atTop (nhds (Xlim ω))) ∧\n (∀ᵐ ω ∂μ, Summable (fun t => W t ω)) := by\n sorry", "doc": "Robbins–Siegmund variant under expectation-level summability and a uniform product bound.\n\nAssumptions:\n- Adaptedness/predictability for `X,Y,Z,W` as in the main theorem\n- Nonnegativity: `0 ≤ X t ω, 0 ≤ Y t ω, 0 ≤ Z t ω, 0 ≤ W t ω`\n- Integrability: `X t, Z t, W t` integrable for all `t`\n- Drift: `μ[X_{t+1} | ℱ_t] ≤ (1+Y_{t+1}) X_t + Z_{t+1} - W_{t+1}` a.e.\n- Expectation summability: `Summable (fun t => ∫ ω, Z t ω ∂μ)`\n- Product bound: `∃ C > 0, ∀ t ω, prodY Y t ω ≤ C`\n\nConclusions:\n- `X t` converges almost surely to a finite limit\n- `∑ W t` is finite almost surely", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:fb0b2b8b1069da839cbe2daef552cc11976bb7d6865e1cabdbefc80b045036ae", "signoffs": [ { "github_user": "GasStationManager", "date": "2026-02-17", "status": "current", "issue": 1, "verdict": "approved", "comment": "S. Gadat, Lecture notes on Stochastic Optimization Algorithms, http://perso.math.univ-toulouse.fr/gadat/files/2012/12/cours_Algo_Stos_M2R5.pdf" } ], "mathlib_conflicts": [], "url": "https://github.com/GasStationManager/ArtificialTheorems/blob/9c0f970db29553e3c32bccb25fd13c3f54dc3a8e" }, { "id": "artificial-theorems:QLS.Stoch.robbinsSiegmund_full", "name": "QLS.Stoch.robbinsSiegmund_full", "kind": "theorem", "origin": "registry", "entry": "artificial-theorems", "entry_name": "ArtificialTheorems", "repository": "https://github.com/GasStationManager/ArtificialTheorems", "commit": "9c0f970db29553e3c32bccb25fd13c3f54dc3a8e", "lean": "leanprover/lean4:v4.27.0", "spec_file": "specs/artificial-theorems/Registry/ArtificialTheorems/Opt/RobbinsSiegmund.lean", "impl_module": "ArtificialTheorems.Opt.RobbinsSiegmund", "lines": [ 75, 113 ], "statement": "theorem robbinsSiegmund_full.{v}\n {Ω : Type v} [m0 : MeasurableSpace Ω]\n (μ : Measure Ω) [IsProbabilityMeasure μ]\n (ℱ : Filtration ℕ m0)\n (V U α β : ℕ → Ω → ℝ)\n -- Adaptedness\n (adapted_V : Adapted ℱ V)\n (adapted_α : Adapted ℱ α)\n (adapted_β : Adapted ℱ β)\n (adapted_U : Adapted ℱ U)\n -- Predictability (X_{n+1} is F_n-measurable)\n (predictable_α : Adapted ℱ fun t => α (t + 1))\n (predictable_β : Adapted ℱ fun t => β (t + 1))\n (predictable_U : Adapted ℱ fun t => U (t + 1))\n -- Nonnegativity\n (hV_nonneg : ∀ t ω, 0 ≤ V t ω)\n (hα_nonneg : ∀ t ω, 0 ≤ α t ω)\n (hβ_nonneg : ∀ t ω, 0 ≤ β t ω)\n (hU_nonneg : ∀ t ω, 0 ≤ U t ω)\n -- Integrability\n (integrable_V : ∀ t, Integrable (V t) μ)\n (integrable_β : ∀ t, Integrable (β t) μ)\n (integrable_U : ∀ t, Integrable (U t) μ)\n -- (ii) Product bound and β summability\n (prod_bound : ∃ C : ℝ, 0 < C ∧ ∀ t ω, prodY α t ω ≤ C)\n (sum_Eβ : Summable (fun t => ∫ ω, β t ω ∂μ))\n -- (iii) Drift inequality\n (condexp_ineq : ∀ t,\n μ[fun ω => V (t + 1) ω | ℱ t]\n ≤ᵐ[μ] fun ω => (1 + α (t + 1) ω) * V t ω + β (t + 1) ω - U (t + 1) ω)\n : -- Conclusions\n -- (a) V_n → V_∞ a.s. with V_∞ ∈ L¹, and sup E[V_n] < ∞\n (∃ Vlim : Ω → ℝ,\n Integrable Vlim μ ∧\n (∀ᵐ ω ∂μ, Tendsto (fun t => V t ω) atTop (nhds (Vlim ω)))) ∧\n (BddAbove (Set.range fun n => ∫ ω, V n ω ∂μ)) ∧\n -- (b) ∑ U_n < ∞ a.s.\n (∀ᵐ ω ∂μ, Summable (fun t => U t ω)) := by\n sorry", "doc": "Full Robbins-Siegmund theorem with L¹ limit and sup E[V] bound.\n\nAssumptions:\n- Adaptedness/predictability for `V, α, β, U`\n- Nonnegativity: `0 ≤ V t ω, 0 ≤ α t ω, 0 ≤ β t ω, 0 ≤ U t ω`\n- Integrability: `V t, β t, U t` integrable for all `t`\n- Product bound: `∃ C > 0, ∀ t ω, prodY α t ω ≤ C`\n- Summability: `Summable (fun t => ∫ ω, β t ω ∂μ)`\n- Drift inequality: `μ[V_{t+1} | ℱ_t] ≤ (1+α_{t+1}) V_t + β_{t+1} - U_{t+1}` a.e.\n\nConclusions:\n- `V_n → V_∞` a.s. with `V_∞ ∈ L¹`\n- `sup E[V_n] < ∞`\n- `∑ U_n < ∞` a.s.", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:66258829305b8150922e40effa5f0ad206d0d4f11765bb6522eb2202db668dce", "signoffs": [ { "github_user": "GasStationManager", "date": "2026-02-17", "status": "current", "issue": 1, "verdict": "approved", "comment": "S. Gadat, Lecture notes on Stochastic Optimization Algorithms, http://perso.math.univ-toulouse.fr/gadat/files/2012/12/cours_Algo_Stos_M2R5.pdf" } ], "mathlib_conflicts": [], "url": "https://github.com/GasStationManager/ArtificialTheorems/blob/9c0f970db29553e3c32bccb25fd13c3f54dc3a8e" }, { "id": "artificial-theorems:convergence_stochastic_gradient_method", "name": "convergence_stochastic_gradient_method", "kind": "theorem", "origin": "registry", "entry": "artificial-theorems", "entry_name": "ArtificialTheorems", "repository": "https://github.com/GasStationManager/ArtificialTheorems", "commit": "9c0f970db29553e3c32bccb25fd13c3f54dc3a8e", "lean": "leanprover/lean4:v4.27.0", "spec_file": "specs/artificial-theorems/Registry/ArtificialTheorems/Opt/SGD.lean", "impl_module": "ArtificialTheorems.Opt.SGD", "lines": [ 79, 94 ], "statement": "theorem convergence_stochastic_gradient_method\n (X : ℕ → Ω → E) (γ : ℕ → ℝ) (h : E → E) (ΔM R : ℕ → Ω → E) (V : E → ℝ) (gradV : E → E)\n (ℱ : Filtration ℕ m0)\n (proc : StochasticAlgorithm X γ h ΔM R)\n (asm : SGD_Convergence_Assumptions μ X γ h ΔM R V gradV ℱ) :\n -- (a) sup E[V(X_n)] < +∞\n (BddAbove (Set.range fun n => ∫ ω, V (X n ω) ∂μ)) ∧\n -- (b) ∑ γ_{n+1}⟨∇V, h⟩(X_n) < +∞ a.s.\n (∀ᵐ ω ∂μ, Summable (fun n => γ (n + 1) * @inner ℝ _ _ (gradV (X n ω)) (h (X n ω)))) ∧\n -- (c) V(X_n) → V_∞ ∈ L¹ a.s.\n (∃ V_inf : Ω → ℝ, Integrable V_inf μ ∧\n ∀ᵐ ω ∂μ, Tendsto (fun n => V (X n ω)) atTop (nhds (V_inf ω))) ∧\n -- (d) X_{n+1} - X_n → 0 a.s. and in L²\n (∀ᵐ ω ∂μ, Tendsto (fun n => X (n + 1) ω - X n ω) atTop (nhds 0)) ∧\n (Tendsto (fun n => ∫ ω, ‖X (n + 1) ω - X n ω‖^2 ∂μ) atTop (nhds 0)) := by\n sorry", "doc": "Theorem 2.3.6: Convergence of the Stochastic Gradient Method.\n\nUnder assumptions (H1) and (H2), one has:\n(a) sup_{n≥0} E[V(X_n)] < +∞\n(b) ∑_{n≥0} γ_{n+1}⟨∇V, h⟩(X_n) < +∞ a.s.\n(c) V(X_n) → V_∞ ∈ L¹ a.s.\n(d) X_n - X_{n-1} → 0 a.s. and in L²", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:d0fadfb9f91bd7033845b0d534fe489d051b6c6a1fec94b536eb712c92c1c01c", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/GasStationManager/ArtificialTheorems/blob/9c0f970db29553e3c32bccb25fd13c3f54dc3a8e" }, { "id": "artificial-theorems:SGDUniqueMin.convergence_simplified", "name": "SGDUniqueMin.convergence_simplified", "kind": "theorem", "origin": "registry", "entry": "artificial-theorems", "entry_name": "ArtificialTheorems", "repository": "https://github.com/GasStationManager/ArtificialTheorems", "commit": "9c0f970db29553e3c32bccb25fd13c3f54dc3a8e", "lean": "leanprover/lean4:v4.27.0", "spec_file": "specs/artificial-theorems/Registry/ArtificialTheorems/Opt/SGDUniqueMin.lean", "impl_module": "ArtificialTheorems.Opt.SGDUniqueMin", "lines": [ 112, 117 ], "statement": "theorem convergence_simplified\n (X : ℕ → Ω → E) (γ : ℕ → ℝ) (h : E → E) (ΔM : ℕ → Ω → E)\n (V : E → ℝ) (gradV : E → E) (ℱ : Filtration ℕ m0) (x_star : E)\n (asm : SimplifiedAssumptions μ X γ h ΔM V gradV ℱ x_star) :\n ∀ᵐ ω ∂μ, Tendsto (fun n => X n ω) atTop (nhds x_star) := by\n sorry", "doc": "Simplified Corollary 2.3.1: Almost sure convergence for unbiased SGD with bounded variance.\n\nUnder the simplified assumptions (unbiased gradients, constant variance bound), we have:\n- X_n → x* almost surely\n\nThe proof is simpler because:\n1. No remainder term R (so the earlier \"line 1375\" gap disappears)\n2. Constant variance ⟹ E[‖∑ γ_k ΔM_k‖²] ≤ σ² ∑ γ_k² < ∞ directly\n3. L²-bounded martingale converges a.s. by Doob's theorem", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:224f7c61562fb8263d474f51bda5ad328d3d6f597b64128d69cf78db4942d960", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/GasStationManager/ArtificialTheorems/blob/9c0f970db29553e3c32bccb25fd13c3f54dc3a8e" }, { "id": "artificial-theorems:VALUE_ITERATION_CONVERGENCE_COMPLETE", "name": "VALUE_ITERATION_CONVERGENCE_COMPLETE", "kind": "theorem", "origin": "registry", "entry": "artificial-theorems", "entry_name": "ArtificialTheorems", "repository": "https://github.com/GasStationManager/ArtificialTheorems", "commit": "9c0f970db29553e3c32bccb25fd13c3f54dc3a8e", "lean": "leanprover/lean4:v4.27.0", "spec_file": "specs/artificial-theorems/Registry/ArtificialTheorems/RL/ValueIterationComplete.lean", "impl_module": "ArtificialTheorems.RL.ValueIterationComplete", "lines": [ 39, 52 ], "statement": "theorem VALUE_ITERATION_CONVERGENCE_COMPLETE (mdp : MDP S A) (γ : Rat)\n (hγ_nonneg : 0 ≤ γ) (hγ_lt : γ < 1) :\n ∃ v_star : S → ℝ,\n -- 1. v_star is the optimal value function (Bellman equation)\n bellmanOperatorReal mdp γ v_star = v_star ∧\n -- 2. Value iteration converges to v_star from any starting point\n (∀ v₀ : S → Rat, Tendsto (fun n => castToReal ((bellmanOperatorRat mdp γ)^[n] v₀)) atTop (𝓝 v_star)) ∧\n -- 3. Geometric convergence with explicit rate\n (∀ v₀ : S → Rat, ∀ n : ℕ,\n dist (castToReal ((bellmanOperatorRat mdp γ)^[n] v₀)) v_star ≤\n dist v₀ (bellmanOperatorRat mdp γ v₀) * γ^n / (1 - γ)) ∧\n -- 4. Uniqueness: any fixed point of the Bellman operator equals v_star\n (∀ v' : S → ℝ, bellmanOperatorReal mdp γ v' = v' → v' = v_star) := by\n sorry", "doc": "**THE MAIN RESULT**: Value iteration converges with all guarantees", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:a24c4794337feb021b922709c94d4cf3e0d6a16245c34b37ad59c5af3feed272", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/GasStationManager/ArtificialTheorems/blob/9c0f970db29553e3c32bccb25fd13c3f54dc3a8e" }, { "id": "artificial-theorems:ApproxValueIterationInt.INT_VALUE_ITERATION_APPROX", "name": "ApproxValueIterationInt.INT_VALUE_ITERATION_APPROX", "kind": "theorem", "origin": "registry", "entry": "artificial-theorems", "entry_name": "ArtificialTheorems", "repository": "https://github.com/GasStationManager/ArtificialTheorems", "commit": "9c0f970db29553e3c32bccb25fd13c3f54dc3a8e", "lean": "leanprover/lean4:v4.27.0", "spec_file": "specs/artificial-theorems/Registry/ArtificialTheorems/RL/ApproxValueIterationInt.lean", "impl_module": "ArtificialTheorems.RL.ApproxValueIterationInt", "lines": [ 59, 67 ], "statement": "theorem INT_VALUE_ITERATION_APPROX\n (mdp : MDP S A) (γ : ℚ) (hγ_nonneg : 0 ≤ γ) (hγ_lt : γ < 1) :\n ∃ v_star : S → ℝ,\n bellmanOperatorReal (S:=S) (A:=A) mdp (γ : ℝ) v_star = v_star ∧\n ∀ (v₀ : S → ℤ) (n : ℕ),\n dist (castZtoR ((bellmanOperatorInt (S:=S) (A:=A) mdp γ)^[n] v₀)) v_star\n ≤ (γ : ℝ)^n * dist (castZtoR v₀) v_star +\n ((1 : ℝ) / 2) / (1 - (γ : ℝ)) := by\n sorry", "doc": "Approximate convergence of integer value iteration.\n\nFor discount 0 ≤ γ < 1, the iterate error is bounded by:\n γ^n * dist(v₀, v*) + (1/2) / (1-γ)", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:2434c8b8c2729171d16c2c999c32573484f64dc587be1e5936fb8fc1f35b39af", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/GasStationManager/ArtificialTheorems/blob/9c0f970db29553e3c32bccb25fd13c3f54dc3a8e" }, { "id": "artificial-theorems:ApproxValueIterationInt.INT_VALUE_ITERATION_EVENTUAL_BALL", "name": "ApproxValueIterationInt.INT_VALUE_ITERATION_EVENTUAL_BALL", "kind": "theorem", "origin": "registry", "entry": "artificial-theorems", "entry_name": "ArtificialTheorems", "repository": "https://github.com/GasStationManager/ArtificialTheorems", "commit": "9c0f970db29553e3c32bccb25fd13c3f54dc3a8e", "lean": "leanprover/lean4:v4.27.0", "spec_file": "specs/artificial-theorems/Registry/ArtificialTheorems/RL/ApproxValueIterationInt.lean", "impl_module": "ArtificialTheorems.RL.ApproxValueIterationInt", "lines": [ 72, 79 ], "statement": "theorem INT_VALUE_ITERATION_EVENTUAL_BALL\n (mdp : MDP S A) (γ : ℚ) (hγ_nonneg : 0 ≤ γ) (hγ_lt : γ < 1)\n (ε : ℝ) (hε : ((1 : ℝ) / 2) / (1 - (γ : ℝ)) < ε) :\n ∃ v_star : S → ℝ,\n bellmanOperatorReal (S:=S) (A:=A) mdp (γ : ℝ) v_star = v_star ∧\n ∀ v₀ : S → ℤ, ∀ᶠ n in atTop,\n dist (castZtoR ((bellmanOperatorInt (S:=S) (A:=A) mdp γ)^[n] v₀)) v_star ≤ ε := by\n sorry", "doc": "Eventual ball inclusion.\n\nFor any ε > (1/2)/(1-γ), the iterates eventually stay within distance ε of v*.", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:ddccb9099c77c89f73a039a1535c68a60fc93350e4e1f631b436830d7e1888c9", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/GasStationManager/ArtificialTheorems/blob/9c0f970db29553e3c32bccb25fd13c3f54dc3a8e" }, { "id": "artificial-theorems:universal_approximation_cybenko", "name": "universal_approximation_cybenko", "kind": "theorem", "origin": "registry", "entry": "artificial-theorems", "entry_name": "ArtificialTheorems", "repository": "https://github.com/GasStationManager/ArtificialTheorems", "commit": "9c0f970db29553e3c32bccb25fd13c3f54dc3a8e", "lean": "leanprover/lean4:v4.27.0", "spec_file": "specs/artificial-theorems/Registry/ArtificialTheorems/Approx/UniversalApprox.lean", "impl_module": "ArtificialTheorems.Approx.UniversalApprox", "lines": [ 44, 52 ], "statement": "theorem universal_approximation_cybenko\n (hJD : HasJordanDecomposition n)\n (σ : ℝ → ℝ) (hσ_cont : Continuous σ) (hσ_sig : IsSigmoidal σ)\n (f : (Fin n → ℝ) → ℝ) (hf_cont : ContinuousOn f (UnitCube n))\n (ε : ℝ) (hε : 0 < ε) :\n ∃ (N : ℕ) (w : Fin N → (Fin n → ℝ)) (b : Fin N → ℝ) (α : Fin N → ℝ),\n ∀ x ∈ UnitCube n,\n |f x - neuralNetFun σ N w b α x| < ε := by\n sorry", "doc": "**Cybenko's Universal Approximation Theorem (1989).**", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:624c726431ffe8ac24fd0b702be3dbb72d103d7d90145a6eee3337b0482c5675", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/GasStationManager/ArtificialTheorems/blob/9c0f970db29553e3c32bccb25fd13c3f54dc3a8e" }, { "id": "stat-learning:GaussianMeasure.stdGaussianPi", "name": "GaussianMeasure.stdGaussianPi", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/GaussianMeasure.lean", "impl_module": "SLT.GaussianMeasure", "lines": [ 14, 15 ], "statement": "noncomputable def stdGaussianPi (n : ℕ) : Measure (Fin n → ℝ) :=\n Measure.pi fun _ : Fin n => gaussianReal 0 1", "doc": "Standard Gaussian product measure: the product of n independent standard Gaussians N(0,1)", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:c6969406214c75ea80270cc8e4012f16414dd4f13c7b0ac3f55710d3cfebbe78", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:GaussianMeasure.stdGaussianE", "name": "GaussianMeasure.stdGaussianE", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/GaussianMeasure.lean", "impl_module": "SLT.GaussianMeasure", "lines": [ 18, 19 ], "statement": "noncomputable def stdGaussianE (n : ℕ) : Measure (EuclideanSpace ℝ (Fin n)) :=\n Measure.map (EuclideanSpace.equiv (Fin n) ℝ).symm (stdGaussianPi n)", "doc": "The standard Gaussian on EuclideanSpace as pushforward of stdGaussianPi via the equivalence.", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:270283eb34f520aa51585284734d5e67f984715851e2ecd82fb90f3fb1bd2f78", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:IsENet", "name": "IsENet", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/CoveringNumber.lean", "impl_module": "SLT.CoveringNumber", "lines": [ 13, 14 ], "statement": "def IsENet {A : Type*} [PseudoMetricSpace A] (t : Finset A) (eps : ℝ) (s : Set A) : Prop :=\n s ⊆ ⋃ x ∈ t, closedBall x eps", "doc": "`t` is an `eps`-net for `s` if every point of `s` lies in a closed ball of radius `eps`\ncentered at some element of `t`.", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:0eb17cca66b05571062342bb25184463d6428116daea40a0aa22105f6e503ce7", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:coveringNumber", "name": "coveringNumber", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/CoveringNumber.lean", "impl_module": "SLT.CoveringNumber", "lines": [ 17, 18 ], "statement": "noncomputable def coveringNumber {A : Type*} [PseudoMetricSpace A] (eps : ℝ) (s : Set A) : WithTop Nat :=\n sInf {n : WithTop Nat | ∃ t : Finset A, IsENet t eps s ∧ (t.card : WithTop Nat) = n}", "doc": "Covering number: the minimal cardinality of a finite `eps`-net, as `WithTop Nat`.", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:bfcb897e02dc18748d41b16d168d1b712f899dd2cecaf7318f3db5c2f6dece51", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:metricEntropyOfNat", "name": "metricEntropyOfNat", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/MetricEntropy.lean", "impl_module": "SLT.MetricEntropy", "lines": [ 32, 33 ], "statement": "def metricEntropyOfNat (n : ℕ) : ℝ :=\n if n ≤ 1 then 0 else Real.log n", "doc": "Helper to compute metric entropy given a natural number.", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:af786dfe49c1674d765f8420662c2ee0b065d7680973bcd4bb51c38dd5fad6cd", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:metricEntropy", "name": "metricEntropy", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/MetricEntropy.lean", "impl_module": "SLT.MetricEntropy", "lines": [ 36, 39 ], "statement": "def metricEntropy (eps : ℝ) (s : Set A) : ℝ :=\n match _h : coveringNumber eps s with\n | ⊤ => 0\n | (n : ℕ) => metricEntropyOfNat n", "doc": "Metric entropy: log of covering number.", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:f3a1d42465927010c9c791bf6eb7379180ad66b40372f501a4d861f870e628b9", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:sqrtEntropy", "name": "sqrtEntropy", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/MetricEntropy.lean", "impl_module": "SLT.MetricEntropy", "lines": [ 42, 43 ], "statement": "def sqrtEntropy (eps : ℝ) (s : Set A) : ℝ :=\n Real.sqrt (metricEntropy eps s)", "doc": "Square root of metric entropy.", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:6efb694da3bb19626a735c08c294010ef0e722cf31b8ba4d4a2c966b7703a0ef", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:dudleyIntegrand", "name": "dudleyIntegrand", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/MetricEntropy.lean", "impl_module": "SLT.MetricEntropy", "lines": [ 46, 47 ], "statement": "def dudleyIntegrand (eps : ℝ) (s : Set A) : ℝ≥0∞ :=\n ENNReal.ofReal (sqrtEntropy eps s)", "doc": "Dudley integrand: √(log N(ε, s)) as ENNReal.", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:84ec0c4922b28bb60bd732efe01002202b19078a497c8ce58beaf26b88452353", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:entropyIntegralENNReal", "name": "entropyIntegralENNReal", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/MetricEntropy.lean", "impl_module": "SLT.MetricEntropy", "lines": [ 50, 51 ], "statement": "def entropyIntegralENNReal (s : Set A) (D : ℝ) : ℝ≥0∞ :=\n ∫⁻ eps in Set.Ioc (0 : ℝ) D, dudleyIntegrand eps s", "doc": "Entropy integral (ENNReal): ∫₀^D √(log N(ε, s)) dε.", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:b85d8e8b840034d3064830f12d943496723dc43aa6fbc082f44eb3b6b302a2dd", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:entropyIntegral", "name": "entropyIntegral", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/MetricEntropy.lean", "impl_module": "SLT.MetricEntropy", "lines": [ 54, 55 ], "statement": "def entropyIntegral (s : Set A) (D : ℝ) : ℝ :=\n (entropyIntegralENNReal s D).toReal", "doc": "Entropy integral (real-valued wrapper).", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:9f975d8b76e9cbe3746aa2998b8c155937d70a9dcebb19e1b3e183e9b6bd009d", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:IsSubGaussianProcess", "name": "IsSubGaussianProcess", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/SubGaussian.lean", "impl_module": "SLT.SubGaussian", "lines": [ 17, 19 ], "statement": "def IsSubGaussianProcess (μ : Measure Ω) (X : A → Ω → ℝ) (σ : ℝ) : Prop :=\n ∀ s t : A, ∀ l : ℝ, μ[fun ω => exp (l * (X s ω - X t ω))] ≤\n exp (l^2 * σ^2 * (dist s t)^2 / 2)", "doc": "A stochastic process X indexed by a metric space is σ-sub-Gaussian if\nthe MGF of increments satisfies E[exp(l(X_s - X_t))] ≤ exp(l²σ²d(s,t)²/2).", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:c0fdd95237411b16ec3370b92d620864348a3010ca4c2cc8aab0b4f57f5088c0", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:EfronSteinApp.CompactlySupportedSmooth", "name": "EfronSteinApp.CompactlySupportedSmooth", "kind": "def", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/EfronSteinApp.lean", "impl_module": "SLT.GaussianPoincare.EfronSteinApp", "lines": [ 15, 16 ], "statement": "def CompactlySupportedSmooth (f : ℝ → ℝ) : Prop :=\n ContDiff ℝ 2 f ∧ HasCompactSupport f", "doc": "A function is compactly supported and smooth (C²)", "informal": "", "informal_source": "", "checks": { "comparator": "not-applicable" }, "verdict_state": "current", "spec_hash": "sha256:d2de3acec8dd9d9f925e6f5a32da99e5f428409d5f0bfb9ef189779a7ad56331", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:GaussianLipConcen.gaussian_lipschitz_concentration", "name": "GaussianLipConcen.gaussian_lipschitz_concentration", "kind": "theorem", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/GaussianLipschitzConcentration.lean", "impl_module": "SLT.GaussianLipConcen", "lines": [ 39, 43 ], "statement": "theorem gaussian_lipschitz_concentration {f : EuclideanSpace ℝ (Fin n) → ℝ} {L : ℝ≥0}\n (hn : 0 < n) (hL : 0 < L) (hf : LipschitzWith L f) (t : ℝ) (ht : 0 < t) :\n let μ := stdGaussianE n\n (μ {x | t ≤ |f x - ∫ y, f y ∂μ|}).toReal ≤ 2 * exp (-t^2 / (2 * (L : ℝ)^2)) := by\n sorry", "doc": "**Gaussian Lipschitz Concentration Inequality**\n\nFor any L-Lipschitz function f on ℝⁿ equipped with the standard Gaussian measure,\nthe probability that f deviates from its mean by more than t is at most\n2 exp(-t² / (2L²)).", "informal": "", "informal_source": "", "checks": { "comparator": "pass" }, "verdict_state": "current", "spec_hash": "sha256:6d4b653df227a944d2de99ff7ed7790b5b02d8eb4d84eb1020ca560be74aab40", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:dudley", "name": "dudley", "kind": "theorem", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/Dudley.lean", "impl_module": "SLT.Dudley", "lines": [ 86, 97 ], "statement": "theorem dudley {μ : Measure Ω} [IsProbabilityMeasure μ]\n {X : A → Ω → ℝ} {σ : ℝ} (hσ : 0 < σ)\n (hX : IsSubGaussianProcess μ X σ)\n {s : Set A} (hs : TotallyBounded s)\n {D : ℝ} (hD : 0 < D) (hdiam : Metric.diam s ≤ D)\n (t₀ : A) (ht₀ : t₀ ∈ s) (hcenter : ∀ ω, X t₀ ω = 0)\n (hX_meas : ∀ t, Measurable (X t))\n (hX_int_exp : ∀ t s : A, ∀ l : ℝ, Integrable (fun ω => Real.exp (l * (X t ω - X s ω))) μ)\n (hfinite : entropyIntegralENNReal s D ≠ ⊤)\n (hcont : ∀ ω, Continuous (fun (t : ↥s) => X t.1 ω)) :\n ∫ ω, ⨆ t ∈ s, X t ω ∂μ ≤ (12 * Real.sqrt 2) * σ * entropyIntegral s D := by\n sorry", "doc": "**Dudley's Entropy Integral Theorem**\n\nFor a σ-sub-Gaussian process X indexed by a totally bounded set s with diameter ≤ D,\nthe expected supremum satisfies:\n E[sup_{t∈s} X_t] ≤ 12√2 · σ · ∫₀^D √(log N(ε, s)) dε", "informal": "", "informal_source": "", "checks": { "comparator": "pass" }, "verdict_state": "current", "spec_hash": "sha256:7507698de20bf960ffd454d224231b2870a71a845d5cec0d511817940ae4fa02", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:efronStein", "name": "efronStein", "kind": "theorem", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/EfronStein.lean", "impl_module": "SLT.EfronStein", "lines": [ 24, 27 ], "statement": "theorem efronStein (f : (Fin n → Ω) → ℝ) (hf : MemLp f 2 (Measure.pi μs)) :\n variance f (Measure.pi μs) ≤\n ∑ i : Fin n, ∫ x, (f x - condExpExceptCoord (μs := μs) i f x)^2 ∂(Measure.pi μs) := by\n sorry", "doc": "**Efron-Stein Inequality**\n\nFor independent random variables X₁,...,Xₙ and a square-integrable function f:\n Var(f) ≤ ∑ᵢ E[(f - E^{(i)}f)²]\nwhere E^{(i)}f is the conditional expectation given all variables except Xᵢ.", "informal": "", "informal_source": "", "checks": { "comparator": "pass" }, "verdict_state": "current", "spec_hash": "sha256:1ef30ab4f8e008e48d503f91fe5de5692eba97503db635d46397a12acbf8ea4f", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "stat-learning:GaussianPoincare.gaussianPoincare", "name": "GaussianPoincare.gaussianPoincare", "kind": "theorem", "origin": "registry", "entry": "stat-learning", "entry_name": "Lean Statistical Learning Theory", "repository": "https://github.com/YuanheZ/lean-stat-learning-theory", "commit": "7b82b1323c80f0c21ca449fd12e1c24315ae9782", "lean": "leanprover/lean4:v4.27.0-rc1", "spec_file": "specs/stat-learning/Registry/StatLearning/GaussianPoincare.lean", "impl_module": "SLT.GaussianPoincare.Limit", "lines": [ 46, 49 ], "statement": "theorem gaussianPoincare {f : ℝ → ℝ} (hf : CompactlySupportedSmooth f) :\n variance (fun x => f x) stdGaussianMeasure ≤\n ∫ x, (deriv f x)^2 ∂stdGaussianMeasure := by\n sorry", "doc": "**Gaussian Poincaré Inequality**\n\nFor any f ∈ C²_c(ℝ) and X ~ N(0,1):\n Var(f(X)) ≤ E[f'(X)²]", "informal": "", "informal_source": "", "checks": { "comparator": "pass" }, "verdict_state": "current", "spec_hash": "sha256:a124288f9993f9f1b0b07b0bf16ece68361b094e592ddc8b9a4c8dff01869804", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/YuanheZ/lean-stat-learning-theory/blob/7b82b1323c80f0c21ca449fd12e1c24315ae9782" }, { "id": "aks:networks_exist", "name": "networks_exist", "kind": "theorem", "origin": "registry", "entry": "aks", "entry_name": "AKS Sorting Networks", "repository": "https://github.com/girving/aks", "commit": "f172ac6c2e46cf30b3aae7e4d19704905133b2bc", "lean": "leanprover/lean4:v4.29.0-rc4", "spec_file": "specs/aks/Registry/AKS/Challenge.lean", "impl_module": "AKS.Seiferas", "lines": [ 64, 66 ], "statement": "theorem networks_exist (n : ℕ) : ∃ net : ComparatorNetwork n, net.Sorts ∧\n net.depth ≤ 141 * 10 ^ 62 * Nat.clog 2 n ∧\n net.size ≤ 705 * 10 ^ 61 * n * Nat.clog 2 n := by sorry", "doc": "Efficient networks exist", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:e6e5c11854ae1257c82446c3ab14b29c854b726f4b46d1705db99abf3b49b088", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/girving/aks/blob/f172ac6c2e46cf30b3aae7e4d19704905133b2bc" }, { "id": "archon-first-proof:Problem4.harmonic_mean_inequality_full", "name": "Problem4.harmonic_mean_inequality_full", "kind": "theorem", "origin": "registry", "entry": "archon-first-proof", "entry_name": "Archon FirstProof Results", "repository": "https://github.com/frenzymath/Archon-FirstProof-Results", "commit": "35550f2bc0a58289bbe2342a64f10a46ada0f52f", "lean": "leanprover/lean4:v4.28.0", "spec_file": "specs/archon-first-proof/Registry/ArchonFirstProof/Problem4.lean", "impl_module": "FirstProof.FirstProof4.Problem4", "lines": [ 133, 139 ], "statement": "theorem harmonic_mean_inequality_full\n (n : ℕ) (hn : 2 ≤ n) (p q : ℝ[X])\n (hp_monic : p.Monic) (hq_monic : q.Monic)\n (hp_deg : p.natDegree = n) (hq_deg : q.natDegree = n)\n (hp_real : ∀ z : ℂ, (p.map (algebraMap ℝ ℂ)).IsRoot z → z.im = 0)\n (hq_real : ∀ z : ℂ, (q.map (algebraMap ℝ ℂ)).IsRoot z → z.im = 0) :\n invPhiN_poly n (polyBoxPlus n p q) ≥ invPhiN_poly n p + invPhiN_poly n q := by sorry", "doc": "**Main Theorem (Problem 4)**: Harmonic-mean inequality for Φₙ under box-plus\n convolution. For monic real-rooted polynomials p, q of degree n ≥ 2:\n 1/Φₙ(p ⊞ₙ q) ≥ 1/Φₙ(p) + 1/Φₙ(q)", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:96d8353b19f18e3c4d7ea42c02c3e1dca50f0ebba7487b4ec225c58deeda575c", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/frenzymath/Archon-FirstProof-Results/blob/35550f2bc0a58289bbe2342a64f10a46ada0f52f" }, { "id": "archon-first-proof:Problem6.exists_eps_light_subset", "name": "Problem6.exists_eps_light_subset", "kind": "theorem", "origin": "registry", "entry": "archon-first-proof", "entry_name": "Archon FirstProof Results", "repository": "https://github.com/frenzymath/Archon-FirstProof-Results", "commit": "35550f2bc0a58289bbe2342a64f10a46ada0f52f", "lean": "leanprover/lean4:v4.28.0", "spec_file": "specs/archon-first-proof/Registry/ArchonFirstProof/Problem6.lean", "impl_module": "FirstProof.FirstProof6.Problem6", "lines": [ 47, 52 ], "statement": "theorem exists_eps_light_subset\n (G : SimpleGraph V) [DecidableRel G.Adj]\n (ε : ℝ) (hε_pos : 0 < ε) (hε_le : ε ≤ 1) :\n ∃ S : Finset V, IsEpsLight G ε S ∧\n ε / 256 * (Fintype.card V : ℝ) ≤\n (S.card : ℝ) := by sorry", "doc": "**Main Theorem (Problem 6)**: For every simple graph G on a finite vertex set V\n and every ε ∈ (0, 1], there exists an ε-light subset S with |S| ≥ ε/256 · |V|.", "informal": "", "informal_source": "", "checks": {}, "verdict_state": "unbound-verdict", "spec_hash": "sha256:ff829a849b969f44b6b365cac5468140d1b298f7d6afd6f06e081e12b827fbfb", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/frenzymath/Archon-FirstProof-Results/blob/35550f2bc0a58289bbe2342a64f10a46ada0f52f" }, { "id": "lean-zip:Zip.Native.Deflate.inflate_deflateRaw", "name": "Zip.Native.Deflate.inflate_deflateRaw", "kind": "theorem", "origin": "registry", "entry": "lean-zip", "entry_name": "lean-zip", "repository": "https://github.com/kim-em/lean-zip", "commit": "e76f0813faa2893f09b83d027e52754041febc35", "lean": "leanprover/lean4:v4.29.1", "spec_file": "specs/lean-zip/Registry/LeanZip/DeflateRoundtrip.lean", "impl_module": "Zip.Spec.DeflateRoundtrip", "lines": [ 26, 29 ], "statement": "theorem inflate_deflateRaw (data : ByteArray) (level : UInt8)\n (maxOutputSize : Nat) (hsize : data.size < maxOutputSize) :\n Zip.Native.Inflate.inflate (deflateRaw data level) maxOutputSize = .ok data := by\n sorry", "doc": "Unified DEFLATE roundtrip: inflate ∘ deflateRaw = identity.\n Generalized to any `maxOutputSize` large enough to hold the input.", "informal": "", "informal_source": "", "checks": { "comparator": "fail", "nanoda": "not-reached" }, "verdict_state": "current", "spec_hash": "sha256:d43548951044d0e6480f217f1fbc0153eb31cd1143c5f38255877aba5d53e8a9", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/kim-em/lean-zip/blob/e76f0813faa2893f09b83d027e52754041febc35" }, { "id": "lean-zip:Zip.Native.Deflate.deflateRaw_pad", "name": "Zip.Native.Deflate.deflateRaw_pad", "kind": "theorem", "origin": "registry", "entry": "lean-zip", "entry_name": "lean-zip", "repository": "https://github.com/kim-em/lean-zip", "commit": "e76f0813faa2893f09b83d027e52754041febc35", "lean": "leanprover/lean4:v4.29.1", "spec_file": "specs/lean-zip/Registry/LeanZip/DeflateRoundtrip.lean", "impl_module": "Zip.Spec.DeflateRoundtrip", "lines": [ 34, 38 ], "statement": "theorem deflateRaw_pad (data : ByteArray) (level : UInt8) :\n ∃ (contentBits padding : List Bool),\n Deflate.Spec.bytesToBits (deflateRaw data level) = contentBits ++ padding ∧\n padding.length < 8 := by\n sorry", "doc": "The output of `deflateRaw` decomposes into content bits plus short padding.\n Needed by `inflateRaw_endPos_ge` to establish that the native decoder\n consumes all of the deflated byte array.", "informal": "", "informal_source": "", "checks": { "comparator": "fail", "nanoda": "not-reached" }, "verdict_state": "current", "spec_hash": "sha256:d25d58945712561ccc60d9b2959f742de00c797ce98e6b28e359872172612fe0", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/kim-em/lean-zip/blob/e76f0813faa2893f09b83d027e52754041febc35" }, { "id": "lean-zip:Zip.Native.Deflate.deflateRaw_goR_pad", "name": "Zip.Native.Deflate.deflateRaw_goR_pad", "kind": "theorem", "origin": "registry", "entry": "lean-zip", "entry_name": "lean-zip", "repository": "https://github.com/kim-em/lean-zip", "commit": "e76f0813faa2893f09b83d027e52754041febc35", "lean": "leanprover/lean4:v4.29.1", "spec_file": "specs/lean-zip/Registry/LeanZip/DeflateRoundtrip.lean", "impl_module": "Zip.Spec.DeflateRoundtrip", "lines": [ 42, 46 ], "statement": "theorem deflateRaw_goR_pad (data : ByteArray) (level : UInt8) :\n ∃ remaining,\n Deflate.Spec.decode.goR (Deflate.Spec.bytesToBits (deflateRaw data level)) []\n = some (data.data.toList, remaining) ∧ remaining.length < 8 := by\n sorry", "doc": "For the encoder's output, `decode.goR` returns a short remaining (< 8 bits).\n Connects encoder structure to decoder bit consumption.", "informal": "", "informal_source": "", "checks": { "comparator": "fail", "nanoda": "not-reached" }, "verdict_state": "current", "spec_hash": "sha256:de258ae6b513e1f24d3165e60b5095a049165cb00b2ac0c6c28e2d99ef20def0", "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/kim-em/lean-zip/blob/e76f0813faa2893f09b83d027e52754041febc35" }, { "id": "leanpool:2-coloring-1-round:Distributed2Coloring.pStar_ge_23879", "name": "Distributed2Coloring.pStar_ge_23879", "kind": "", "origin": "overlay:leanpool", "entry": "2-coloring-1-round", "entry_name": "2-Coloring Cycles in One Round", "repository": "https://github.com/suomela/2-coloring-1-round", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.TwoColoringOneRound", "lines": null, "statement": "", "doc": "Formalizes certified upper and lower bounds for one-round randomized algorithms that 2-color cycles.", "informal": "Proves that every one-round classical algorithm has monochromatic-edge probability at least 0.23879.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/TwoColoringOneRound", "classification": { "msc2020": [ "68W15", "05C15" ], "tags": [ "distributed-computing", "graph-coloring", "randomized-algorithms", "formal-verification" ], "branch": "distributed computing" } }, { "id": "leanpool:2-coloring-1-round:Distributed2Coloring.pStar_lt_24118", "name": "Distributed2Coloring.pStar_lt_24118", "kind": "", "origin": "overlay:leanpool", "entry": "2-coloring-1-round", "entry_name": "2-Coloring Cycles in One Round", "repository": "https://github.com/suomela/2-coloring-1-round", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.TwoColoringOneRound", "lines": null, "statement": "", "doc": "Formalizes certified upper and lower bounds for one-round randomized algorithms that 2-color cycles.", "informal": "Proves that an explicit one-round algorithm attains monochromatic-edge probability strictly below 0.24118.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/TwoColoringOneRound", "classification": { "msc2020": [ "68W15", "05C15" ], "tags": [ "distributed-computing", "graph-coloring", "randomized-algorithms", "formal-verification" ], "branch": "distributed computing" } }, { "id": "leanpool:a-formalization-of-borel-determinacy-in-lean:GaleStewartGame.borel_determinacy", "name": "GaleStewartGame.borel_determinacy", "kind": "", "origin": "overlay:leanpool", "entry": "a-formalization-of-borel-determinacy-in-lean", "entry_name": "A formalization of Borel determinacy in Lean", "repository": "https://github.com/sven-manthe/A-formalization-of-Borel-determinacy-in-Lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.AFormalizationOfBorelDeterminacyInLean", "lines": null, "statement": "", "doc": "Formalizes Martin's theorem that every Borel Gale-Stewart game is determined.", "informal": "Proves that every Borel Gale-Stewart game on a nonempty discrete alphabet is determined.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/AFormalizationOfBorelDeterminacyInLean", "classification": { "msc2020": [ "03E15", "54H05", "91A44" ], "tags": [ "descriptive-set-theory", "game-theory", "determinacy" ], "branch": "descriptive set theory" } }, { "id": "leanpool:a-formalization-of-borel-determinacy-in-lean:GaleStewartGame.Games.borel_determinacy", "name": "GaleStewartGame.Games.borel_determinacy", "kind": "", "origin": "overlay:leanpool", "entry": "a-formalization-of-borel-determinacy-in-lean", "entry_name": "A formalization of Borel determinacy in Lean", "repository": "https://github.com/sven-manthe/A-formalization-of-Borel-determinacy-in-Lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.AFormalizationOfBorelDeterminacyInLean", "lines": null, "statement": "", "doc": "Formalizes Martin's theorem that every Borel Gale-Stewart game is determined.", "informal": "Proves Borel determinacy for the bundled game representation used in the formalization.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/AFormalizationOfBorelDeterminacyInLean", "classification": { "msc2020": [ "03E15", "54H05", "91A44" ], "tags": [ "descriptive-set-theory", "game-theory", "determinacy" ], "branch": "descriptive set theory" } }, { "id": "leanpool:abc-exceptions:abcConjecture_iff_countTriples", "name": "abcConjecture_iff_countTriples", "kind": "", "origin": "overlay:leanpool", "entry": "abc-exceptions", "entry_name": "Exceptional Set in the abc Conjecture", "repository": "https://github.com/b-mehta/ABC-Exceptions", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ABCExceptions", "lines": null, "statement": "", "doc": "Formalizes reductions and optimization bounds for a power-saving estimate on exceptional triples in the abc conjecture.", "informal": "Recasts the abc conjecture as a uniform big-O bound for bounded exceptional triples.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/ABCExceptions", "classification": { "msc2020": [ "11D75", "11N37" ], "tags": [ "number-theory", "analytic-number-theory", "abc-conjecture" ], "branch": "analytic number theory" } }, { "id": "leanpool:abc-exceptions:thm_4_point_3", "name": "thm_4_point_3", "kind": "", "origin": "overlay:leanpool", "entry": "abc-exceptions", "entry_name": "Exceptional Set in the abc Conjecture", "repository": "https://github.com/b-mehta/ABC-Exceptions", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ABCExceptions", "lines": null, "statement": "", "doc": "Formalizes reductions and optimization bounds for a power-saving estimate on exceptional triples in the abc conjecture.", "informal": "Derives the Section 4 optimization bound from the determinant, geometry, and Fourier hypotheses.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/ABCExceptions", "classification": { "msc2020": [ "11D75", "11N37" ], "tags": [ "number-theory", "analytic-number-theory", "abc-conjecture" ], "branch": "analytic number theory" } }, { "id": "leanpool:abc-exceptions:refinedCountTriplesStar_isBigO_B", "name": "refinedCountTriplesStar_isBigO_B", "kind": "", "origin": "overlay:leanpool", "entry": "abc-exceptions", "entry_name": "Exceptional Set in the abc Conjecture", "repository": "https://github.com/b-mehta/ABC-Exceptions", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ABCExceptions", "lines": null, "statement": "", "doc": "Formalizes reductions and optimization bounds for a power-saving estimate on exceptional triples in the abc conjecture.", "informal": "Bounds the refined exceptional-triple count by the dyadic structured-box supremum.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/ABCExceptions", "classification": { "msc2020": [ "11D75", "11N37" ], "tags": [ "number-theory", "analytic-number-theory", "abc-conjecture" ], "branch": "analytic number theory" } }, { "id": "leanpool:agree-to-disagree:AgreeToDisagree.agreeToDisagree", "name": "AgreeToDisagree.agreeToDisagree", "kind": "", "origin": "overlay:leanpool", "entry": "agree-to-disagree", "entry_name": "Aumann's Agreement Theorem", "repository": 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"llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/AgreeToDisagree", "classification": { "msc2020": [ "60A10", "91A40" ], "tags": [ "probability", "game-theory", "epistemic-logic" ], "branch": "probability theory" } }, { "id": "leanpool:agree-to-disagree:AgreeToDisagree.agreeToDisagree'", "name": "AgreeToDisagree.agreeToDisagree'", "kind": "", "origin": "overlay:leanpool", "entry": "agree-to-disagree", "entry_name": "Aumann's Agreement Theorem", "repository": "https://github.com/AxiomMath/AgreeToDisagree", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.AgreeToDisagree", "lines": null, "statement": "", "doc": "Formalizes Aumann's agreement theorem for information partitions and a common-prior setting, together with a p-belief variant bounding posterior disagreement by 2 * (1 - p).", "informal": "Generalizes the agreement theorem to a family of agents indexed by an arbitrary type.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/AgreeToDisagree", "classification": { "msc2020": [ "60A10", "91A40" ], "tags": [ "probability", "game-theory", "epistemic-logic" ], "branch": "probability theory" } }, { "id": "leanpool:aharonikorman:LeanPool.AharoniKorman.aharoni_korman_false", "name": "LeanPool.AharoniKorman.aharoni_korman_false", "kind": "", "origin": "overlay:leanpool", "entry": "aharonikorman", "entry_name": "Disproof of the Aharoni-Korman Conjecture", "repository": "https://github.com/b-mehta/AharoniKorman", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.AharoniKorman", "lines": null, "statement": "", "doc": "Formalizes Hollom's counterexample to the Aharoni-Korman fishbone conjecture for partially ordered sets.", "informal": "Proves that not every partially ordered set has either an infinite antichain or a chain meeting every antichain in a partition.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/AharoniKorman", "classification": { "msc2020": [ "06A06", "06A07" ], "tags": [ "order-theory", "combinatorics", "partial-orders" ], "branch": "order theory" } }, { "id": "leanpool:anderson-conjecture:anderson_main_theorem", "name": "anderson_main_theorem", "kind": "", "origin": "overlay:leanpool", "entry": "anderson-conjecture", "entry_name": "The Anderson Conjecture: A Weakly Quasi-Complete Ring Need Not Be Quasi-Complete", "repository": 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"leanpool:apportionment:Apportionment.balinski_young", "name": "Apportionment.balinski_young", "kind": "", "origin": "overlay:leanpool", "entry": "apportionment", "entry_name": "Apportionmentlib", "repository": "https://github.com/mdbrnowski/apportionmentlib", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Apportionment", "lines": null, "statement": "", "doc": "Formalizes apportionment rules and proves the Balinski-Young impossibility theorem: no anonymous quota rule can be population-monotone.", "informal": "Proves that no anonymous quota apportionment rule can satisfy population monotonicity.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Apportionment", "classification": { "msc2020": [ "91B12", "91B14" ], "tags": [ "social-choice-theory", "combinatorics" ], "branch": "social choice theory" } }, { "id": "leanpool:archon-firstproof-results:Problem4.harmonic_mean_inequality_full", "name": "Problem4.harmonic_mean_inequality_full", "kind": "", "origin": "overlay:leanpool", "entry": "archon-firstproof-results", "entry_name": "Archon-FirstProof-Results", "repository": "https://github.com/frenzymath/Archon-FirstProof-Results", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ArchonFirstProofResults", "lines": null, "statement": "", "doc": "Collects formal solutions from the Archon FirstProof benchmark, including polynomial and graph-theoretic results.", "informal": "Proves the benchmark harmonic-mean inequality for real-rooted polynomial transforms.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, 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"informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/ArchonFirstProofResults", "classification": { "msc2020": [ "26D15", "05C50", "15A42" ], "tags": [ "polynomials", "analysis", "combinatorics", "linear-algebra", "graph-theory" ], "branch": "problem-solving mathematics" } }, { "id": "leanpool:artin-wedderburn:LeanPool.ArtinWedderburn.ArtinWedderburnForPrime", "name": "LeanPool.ArtinWedderburn.ArtinWedderburnForPrime", "kind": "", "origin": "overlay:leanpool", "entry": "artin-wedderburn", "entry_name": "Artin-Wedderburn Theorem", "repository": "https://github.com/JobPetrovcic/ArtinWedderburn", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ArtinWedderburn", "lines": null, "statement": "", "doc": "Formalizes 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"url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Brouwer", "classification": { "msc2020": [ "91A06", "91A10", "47H10" ], "tags": [ "game-theory", "nash-equilibrium", "brouwer-fixed-point", "scarf-lemma", "simplex" ], "branch": "game theory" } }, { "id": "leanpool:bruhat-tits:BruhatTits.BTtree", "name": "BruhatTits.BTtree", "kind": "", "origin": "overlay:leanpool", "entry": "bruhat-tits", "entry_name": "Formalisation of the Bruhat-Tits Tree", "repository": "https://github.com/chrisflav/bruhat-tits", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.BruhatTits", "lines": null, "statement": "", "doc": "Formalizes the Bruhat-Tits tree for GL₂ over the fraction field of a discrete valuation ring.", "informal": "Proves that the Bruhat-Tits graph is a tree.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/BruhatTits", "classification": { "msc2020": [ "20E42", "05C25", "13H05" ], "tags": [ "algebraic-geometry", "graph-theory", "discrete-valuation-rings" ], "branch": "algebraic geometry" } }, { "id": "leanpool:bruhat-tits:BruhatTits.BTlaplace_surjective", "name": "BruhatTits.BTlaplace_surjective", "kind": "", "origin": "overlay:leanpool", "entry": "bruhat-tits", "entry_name": "Formalisation of the Bruhat-Tits Tree", "repository": "https://github.com/chrisflav/bruhat-tits", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.BruhatTits", "lines": null, "statement": "", "doc": "Formalizes the Bruhat-Tits tree for GL₂ over the fraction field of a discrete valuation ring.", "informal": "Proves surjectivity of the Bruhat-Tits Laplacian on harmonic cochains.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/BruhatTits", "classification": { "msc2020": [ "20E42", "05C25", "13H05" ], "tags": [ "algebraic-geometry", "graph-theory", "discrete-valuation-rings" ], "branch": "algebraic geometry" } }, { "id": "leanpool:bruhat-tits:BruhatTits.btgraph_regular", "name": "BruhatTits.btgraph_regular", "kind": "", "origin": "overlay:leanpool", "entry": "bruhat-tits", "entry_name": "Formalisation of the Bruhat-Tits Tree", "repository": "https://github.com/chrisflav/bruhat-tits", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.BruhatTits", "lines": null, "statement": "", "doc": "Formalizes the Bruhat-Tits tree for GL₂ over the fraction field of a discrete valuation ring.", "informal": "Proves regularity of the Bruhat-Tits graph when the residue field is finite.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/BruhatTits", "classification": { "msc2020": [ "20E42", "05C25", "13H05" ], "tags": [ "algebraic-geometry", "graph-theory", "discrete-valuation-rings" ], "branch": "algebraic geometry" } }, { "id": "leanpool:burkholder:MeasureTheory.Lp_Burkholder_inequality_martingaleTransform", "name": "MeasureTheory.Lp_Burkholder_inequality_martingaleTransform", "kind": "", "origin": "overlay:leanpool", "entry": "burkholder", "entry_name": "Burkholder Martingale Transform Inequality", "repository": "https://github.com/SmaniaD/Burkholder", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Burkholder", "lines": null, "statement": "", "doc": "Formalizes Burkholder's sharp Lp inequality for real-valued martingale transforms: if `f` is a discrete martingale and `w` a predictable multiplier bounded by `1`, then the Lp norm of the transform `w ⋆ₘ f` is at most `(pStar p - 1)` times the Lp norm of `f`, where `pStar p` is the maximum of `p` and its conjugate exponent. 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The proof builds the Burkholder special function as a majorant of the relevant payoff in each exponent regime (`1 < p < 2`, `p = 2`, `p > 2`).", "informal": "The martingale transform of a martingale by a strongly predictable bounded multiplier is itself a martingale.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Burkholder", "classification": { "msc2020": [ "60G42" ], "tags": [ "probability", "martingales", "burkholder-inequality" ], "branch": "probability" } }, { "id": "leanpool:burkholder:Majorants.exists_majorant_p_g_1", "name": "Majorants.exists_majorant_p_g_1", "kind": "", "origin": "overlay:leanpool", "entry": "burkholder", "entry_name": "Burkholder Martingale Transform Inequality", "repository": "https://github.com/SmaniaD/Burkholder", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Burkholder", "lines": null, "statement": "", "doc": "Formalizes Burkholder's sharp Lp inequality for real-valued martingale transforms: if `f` is a discrete martingale and `w` a predictable multiplier bounded by `1`, then the Lp norm of the transform `w ⋆ₘ f` is at most `(pStar p - 1)` times the Lp norm of `f`, where `pStar p` is the maximum of `p` and its conjugate exponent. 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"cencov-petz", "entry_name": "Finite Čencov-Petz Uniqueness", "repository": "https://github.com/abenenson/cencov-petz", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.CencovPetz", "lines": null, "statement": "", "doc": "Formalizes the finite/discrete Čencov-Petz uniqueness theorem, showing that any continuous metric family on finite probability simplexes that is monotone under Markov morphisms is a scalar multiple of the Fisher information metric.", "informal": "Proves that every continuous monotone metric family on a finite probability simplex is a scalar multiple of the Fisher information metric.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/CencovPetz", "classification": { "msc2020": [ "62B10", 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metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/CencovPetz", "classification": { "msc2020": [ "62B10", "53C21" ], "tags": [ "information-geometry", "fisher-information", "markov-morphisms", "finite-simplex" ], "branch": "information geometry" } }, { "id": "leanpool:cencov-petz:LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_isSplitRepresentable", "name": "LeanPool.CencovPetz.MonotoneMetricFamily.eq_smul_fisher_of_isSplitRepresentable", "kind": "", "origin": "overlay:leanpool", "entry": "cencov-petz", "entry_name": "Finite Čencov-Petz Uniqueness", "repository": "https://github.com/abenenson/cencov-petz", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.CencovPetz", "lines": null, "statement": "", "doc": "Formalizes the finite/discrete Čencov-Petz uniqueness theorem, showing that any continuous metric family on finite probability simplexes that is monotone under Markov morphisms is a scalar multiple of the Fisher information metric.", "informal": "Proves the scalar Fisher-form identity for simplex points represented as pushforwards of uniform distributions under splitting maps.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/CencovPetz", "classification": { "msc2020": [ "62B10", "53C21" ], "tags": [ "information-geometry", "fisher-information", "markov-morphisms", "finite-simplex" ], "branch": "information geometry" } }, { "id": "leanpool:cencov-petz:LeanPool.CencovPetz.MonotoneMetricFamily.uniformScalar_eq_uniformScalar_two", "name": 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"https://github.com/Vilin97/lean-pool/tree/main/LeanPool/CencovPetz", "classification": { "msc2020": [ "62B10", "53C21" ], "tags": [ "information-geometry", "fisher-information", "markov-morphisms", "finite-simplex" ], "branch": "information geometry" } }, { "id": "leanpool:cencov-petz:LeanPool.CencovPetz.TangentFin.Bilin.B_eq_smul_fisherBilin_uniform", "name": "LeanPool.CencovPetz.TangentFin.Bilin.B_eq_smul_fisherBilin_uniform", "kind": "", "origin": "overlay:leanpool", "entry": "cencov-petz", "entry_name": "Finite Čencov-Petz Uniqueness", "repository": "https://github.com/abenenson/cencov-petz", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.CencovPetz", "lines": null, "statement": "", "doc": "Formalizes the finite/discrete Čencov-Petz uniqueness theorem, showing that any continuous metric family on finite probability simplexes that is monotone under Markov morphisms is a scalar multiple of the Fisher information metric.", "informal": "Proves that the permutation-invariant bilinear form at a uniform finite simplex point is a scalar multiple of the Fisher bilinear form.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/CencovPetz", "classification": { "msc2020": [ "62B10", "53C21" ], "tags": [ "information-geometry", "fisher-information", "markov-morphisms", "finite-simplex" ], "branch": "information geometry" } }, { "id": "leanpool:channel-capacity:ChannelCapacity.exists_unique_capacity_achieving_prior_of_finite", "name": "ChannelCapacity.exists_unique_capacity_achieving_prior_of_finite", "kind": "", "origin": "overlay:leanpool", "entry": "channel-capacity", "entry_name": "Uniqueness of Shannon Capacity-Achieving Priors", "repository": "https://github.com/abenenson/channel-capacity", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ChannelCapacity", "lines": null, "statement": "", "doc": "Formalizes uniqueness of the capacity-achieving input distribution for a Markov-kernel channel in three aligned layers: a finite-alphabet theorem (compactness, continuity, and strict concavity of mutual information on Mathlib's weak topology for `ProbabilityMeasure`), a concrete discharged theorem for compact-Polish channels with a common strictly positive jointly continuous density, and a general measure-theoretic packaging that isolates the abstract well-conditioning hypotheses, plus a counterexample showing pairwise-distinct rows do not give an injective prior pushforward (the non-degeneracy required for uniqueness).", "informal": "For a Markov kernel between finite alphabets (nonempty input) whose rows are linearly independent (`Kernel.RowMatrixFullRank`), a capacity-achieving input prior exists and is unique.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/ChannelCapacity", "classification": { "msc2020": [ "94A17", "94A15", "60A10" ], "tags": [ "information-theory", "channel-capacity", "mutual-information", "kullback-leibler", "markov-kernel" ], "branch": "information theory" } }, { "id": "leanpool:channel-capacity:ChannelCapacity.exists_unique_capacity_achieving_prior_discharged", "name": "ChannelCapacity.exists_unique_capacity_achieving_prior_discharged", "kind": "", "origin": "overlay:leanpool", "entry": "channel-capacity", "entry_name": "Uniqueness of Shannon Capacity-Achieving Priors", "repository": "https://github.com/abenenson/channel-capacity", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ChannelCapacity", "lines": null, "statement": "", "doc": "Formalizes uniqueness of the capacity-achieving input distribution for a Markov-kernel channel in three aligned layers: a finite-alphabet theorem (compactness, continuity, and strict concavity of mutual information on Mathlib's weak topology for `ProbabilityMeasure`), a concrete discharged theorem for compact-Polish channels with a common strictly positive jointly continuous density, and a general measure-theoretic packaging that isolates the abstract well-conditioning hypotheses, plus a counterexample showing pairwise-distinct rows do not give an injective prior pushforward (the non-degeneracy required for uniqueness).", "informal": "For compact-Polish channels with a common strictly positive jointly continuous density, given injectivity of the prior-pushforward map and upper semicontinuity of mutual information (the latter carried as a field of `Kernel.ContinuousPositiveDensity`, not derived), the capacity-achieving prior is unique.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/ChannelCapacity", "classification": { "msc2020": [ "94A17", "94A15", "60A10" ], "tags": [ "information-theory", "channel-capacity", "mutual-information", "kullback-leibler", "markov-kernel" ], "branch": "information theory" } }, { "id": "leanpool:channel-capacity:ChannelCapacity.exists_unique_capacity_achieving_prior", "name": "ChannelCapacity.exists_unique_capacity_achieving_prior", "kind": "", "origin": "overlay:leanpool", "entry": "channel-capacity", "entry_name": "Uniqueness of Shannon Capacity-Achieving Priors", "repository": "https://github.com/abenenson/channel-capacity", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ChannelCapacity", "lines": null, "statement": "", "doc": "Formalizes uniqueness of the capacity-achieving input distribution for a Markov-kernel channel in three aligned layers: a finite-alphabet theorem (compactness, continuity, and strict concavity of mutual information on Mathlib's weak topology for `ProbabilityMeasure`), a concrete discharged theorem for compact-Polish channels with a common strictly positive jointly continuous density, and a general measure-theoretic packaging that isolates the abstract well-conditioning hypotheses, plus a counterexample showing pairwise-distinct rows do not give an injective prior pushforward (the non-degeneracy required for uniqueness).", "informal": "General measure-theoretic packaging: under abstract well-conditioning hypotheses, the capacity-achieving prior is unique.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/ChannelCapacity", "classification": { "msc2020": [ "94A17", "94A15", "60A10" ], "tags": [ "information-theory", "channel-capacity", "mutual-information", "kullback-leibler", "markov-kernel" ], "branch": "information theory" } }, { "id": "leanpool:chudnovsky:Chudnovsky.chudnovskySum_eq_pi_inv", "name": "Chudnovsky.chudnovskySum_eq_pi_inv", "kind": "", "origin": "overlay:leanpool", "entry": "chudnovsky", "entry_name": "A Detailed Proof of the Chudnovsky Formula", "repository": "https://github.com/ldct/lean-eval-chudnovsky", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Chudnovsky", "lines": null, "statement": "", "doc": "Formalizes the Chudnovsky brothers' rapidly convergent series for 1/π, proving Mathlib's `chudnovskySum = π⁻¹`. Follows Milla's elementary complex-analytic proof: Weierstrass and Eisenstein theory, Liouville and parallelogram-residue calculus, the Kummer hypergeometric bridge and the main modular identity relating the j-invariant to the lattice periods, Baker's division-value machinery, and the singular-moduli inputs at the CM point τ = (1 + √-163)/2 — integrality of j(τ) via modular polynomials and Kronecker's congruence, and rationality of the Weierstrass s₂ invariant via Masser's Appendix A.1.", "informal": "Proves the Chudnovsky series identity, i.e. Mathlib's `chudnovskySum = π⁻¹`: the series (12 / 640320^(3/2)) · ∑ₙ (-1)ⁿ (6n)! (545140134 n + 13591409) / ((3n)! 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Kronecker symbol and the Kodaira type enumeration.", "informal": "Proves that the explicitly constructed singular point of a singular model over a perfect field is indeed a singular point (Connell, Proposition 1.5.4).", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/EcTateLean", "classification": { "msc2020": [ "11G05", "11G07", "14H52" ], "tags": [ "number-theory", "elliptic-curves", "algebraic-geometry" ], "branch": "number theory" } }, { "id": "leanpool:ec-tate-lean:Model.discr_identity", "name": "Model.discr_identity", "kind": "", "origin": "overlay:leanpool", "entry": "ec-tate-lean", "entry_name": "Weierstrass models and singular points for Tate's algorithm", "repository": "https://github.com/KisaraBlue/ec-tate-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.EcTateLean", "lines": null, "statement": "", "doc": "Formalizes Weierstrass models of elliptic curves over a commutative ring, their b- and c-invariants and discriminant, the change-of-coordinates group action, the discriminant's membership in the Jacobian ideal, and the construction of the singular point of a singular model over a perfect field, together with the Kronecker symbol and the Kodaira type enumeration.", "informal": "Proves the identity relating the discriminant to the c-invariants, 1728 * discr = c4^3 - c6^2.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/EcTateLean", "classification": { "msc2020": [ "11G05", "11G07", "14H52" ], "tags": [ "number-theory", "elliptic-curves", "algebraic-geometry" ], "branch": "number theory" } }, { "id": "leanpool:ec-tate-lean:Model.rst_discr", "name": "Model.rst_discr", "kind": "", "origin": "overlay:leanpool", "entry": "ec-tate-lean", "entry_name": "Weierstrass models and singular points for Tate's algorithm", "repository": "https://github.com/KisaraBlue/ec-tate-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.EcTateLean", "lines": null, "statement": "", "doc": "Formalizes Weierstrass models of elliptic curves over a commutative ring, their b- and c-invariants and discriminant, the change-of-coordinates group action, the discriminant's membership in the Jacobian ideal, and the construction of the singular point of a singular model over a perfect field, together with the Kronecker symbol and the Kodaira type enumeration.", "informal": "Proves that the discriminant is invariant under the rst change-of-coordinates transformation.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/EcTateLean", "classification": { "msc2020": [ "11G05", "11G07", "14H52" ], "tags": [ "number-theory", "elliptic-curves", "algebraic-geometry" ], "branch": "number theory" } }, { "id": "leanpool:ec-tate-lean:Model.discr_in_jacobian_ideal", "name": "Model.discr_in_jacobian_ideal", "kind": "", "origin": "overlay:leanpool", "entry": "ec-tate-lean", "entry_name": "Weierstrass models and singular points for Tate's algorithm", "repository": "https://github.com/KisaraBlue/ec-tate-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.EcTateLean", "lines": null, "statement": "", "doc": "Formalizes Weierstrass models of elliptic curves over a commutative ring, their b- and c-invariants and discriminant, the change-of-coordinates group action, the discriminant's membership in the Jacobian ideal, and the construction of the singular point of a singular model over a perfect field, together with the Kronecker symbol and the Kodaira type enumeration.", "informal": "Expresses the discriminant as an explicit combination of the Weierstrass polynomial and its partial derivatives, witnessing membership in the Jacobian ideal.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/EcTateLean", "classification": { "msc2020": [ "11G05", "11G07", "14H52" ], "tags": [ "number-theory", "elliptic-curves", "algebraic-geometry" ], "branch": "number theory" } }, { "id": "leanpool:ec-tate-lean:Model.discr_eq_zero_of_singular", "name": "Model.discr_eq_zero_of_singular", "kind": "", "origin": "overlay:leanpool", "entry": "ec-tate-lean", "entry_name": "Weierstrass models and singular points for Tate's algorithm", "repository": "https://github.com/KisaraBlue/ec-tate-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.EcTateLean", "lines": null, "statement": "", "doc": "Formalizes Weierstrass models of elliptic curves over a commutative ring, their b- and c-invariants and discriminant, the change-of-coordinates group action, the discriminant's membership in the Jacobian ideal, and the construction of the singular point of a singular model over a perfect field, together with the Kronecker symbol and the Kodaira type enumeration.", "informal": "Proves that a model with a singular point has vanishing discriminant.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/EcTateLean", "classification": { "msc2020": [ "11G05", "11G07", "14H52" ], "tags": [ "number-theory", "elliptic-curves", "algebraic-geometry" ], "branch": "number theory" } }, { "id": "leanpool:ec-tate-lean:Model.Field.move_singular_point_to_origin", "name": "Model.Field.move_singular_point_to_origin", "kind": "", "origin": "overlay:leanpool", "entry": "ec-tate-lean", "entry_name": "Weierstrass models and singular points for Tate's algorithm", "repository": "https://github.com/KisaraBlue/ec-tate-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.EcTateLean", "lines": null, "statement": "", "doc": "Formalizes Weierstrass models of elliptic curves over a commutative ring, their b- and c-invariants and discriminant, the change-of-coordinates group action, the discriminant's membership in the Jacobian ideal, and the construction of the singular point of a singular model over a perfect field, together with the Kronecker symbol and the Kodaira type enumeration.", "informal": "Proves that the change of coordinates moving the singular point places it at the origin.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/EcTateLean", "classification": { "msc2020": [ "11G05", "11G07", "14H52" ], "tags": [ "number-theory", "elliptic-curves", "algebraic-geometry" ], "branch": "number theory" } }, { "id": "leanpool:ec-tate-lean:Kodaira", "name": "Kodaira", "kind": "", "origin": "overlay:leanpool", "entry": "ec-tate-lean", "entry_name": "Weierstrass models and singular points for Tate's algorithm", "repository": "https://github.com/KisaraBlue/ec-tate-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.EcTateLean", "lines": null, "statement": "", "doc": "Formalizes Weierstrass models of elliptic curves over a commutative ring, their b- and c-invariants and discriminant, the change-of-coordinates group action, the discriminant's membership in the Jacobian ideal, and the construction of the singular point of a singular model over a perfect field, together with the Kronecker symbol and the Kodaira type enumeration.", "informal": "Enumerates the Kodaira reduction types, including the extra types arising over imperfect residue fields of characteristic 2 or 3.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/EcTateLean", "classification": { "msc2020": [ "11G05", "11G07", "14H52" ], "tags": [ "number-theory", "elliptic-curves", "algebraic-geometry" ], "branch": "number theory" } }, { "id": "leanpool:egrs75:Egrs75.MuFinish.egrs_two_prime_mu", "name": "Egrs75.MuFinish.egrs_two_prime_mu", "kind": "", "origin": "overlay:leanpool", "entry": "egrs75", "entry_name": "The Erdős–Graham–Ruzsa–Straus two-prime theorem", "repository": "https://github.com/lyfar/egrs75-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Egrs75", "lines": null, "statement": "", "doc": "A complete machine-checked proof that for any two distinct odd primes p and q there are infinitely many n with binom(2n,n) coprime to pq, including the iterated carry-controlled digit-clearing step that Bloom and Croot defer to the 1975 paper, via a per-move lexicographic measure, an elementary staircase lemma, and a Dirichlet-approximation seed.", "informal": "Proves that for distinct odd primes p and q there are infinitely many n with p and q both not dividing the central binomial coefficient binom(2n,n), via the mu-measure descent.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Egrs75", "classification": { "msc2020": [ "11A63", "11B65" ], "tags": [ "number-theory", "central-binomial-coefficients", "digit-representations", "erdos-problems" ], "branch": "number theory" } }, { "id": "leanpool:egrs75:Egrs75.Finish.egrs_two_prime_finish", "name": "Egrs75.Finish.egrs_two_prime_finish", "kind": "", "origin": "overlay:leanpool", "entry": "egrs75", "entry_name": "The Erdős–Graham–Ruzsa–Straus two-prime theorem", "repository": "https://github.com/lyfar/egrs75-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Egrs75", "lines": null, "statement": "", "doc": "A complete machine-checked proof that for any two distinct odd primes p and q there are infinitely many n with binom(2n,n) coprime to pq, including the iterated carry-controlled digit-clearing step that Bloom and Croot defer to the 1975 paper, via a per-move lexicographic measure, an elementary staircase lemma, and a Dirichlet-approximation seed.", "informal": "Reassembles the same theorem via bad-bit-mask strong induction, with the low-case clearing discharged by Egrs75.MuFinish.egrs_clearing_low_mu.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Egrs75", "classification": { "msc2020": [ "11A63", "11B65" ], "tags": [ "number-theory", "central-binomial-coefficients", "digit-representations", "erdos-problems" ], "branch": "number theory" } }, { "id": "leanpool:egrs75:Egrs75.not_dvd_centralBinom_iff_lowDigits", "name": "Egrs75.not_dvd_centralBinom_iff_lowDigits", "kind": "", "origin": "overlay:leanpool", "entry": "egrs75", "entry_name": "The Erdős–Graham–Ruzsa–Straus two-prime theorem", "repository": "https://github.com/lyfar/egrs75-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Egrs75", "lines": null, "statement": "", "doc": "A complete machine-checked proof that for any two distinct odd primes p and q there are infinitely many n with binom(2n,n) coprime to pq, including the iterated carry-controlled digit-clearing step that Bloom and Croot defer to the 1975 paper, via a per-move lexicographic measure, an elementary staircase lemma, and a Dirichlet-approximation seed.", "informal": "The Kummer bridge, stating that an odd prime p does not divide binom(2n,n) exactly when every base-p digit of n is at most (p-1)/2.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Egrs75", "classification": { "msc2020": [ "11A63", "11B65" ], "tags": [ "number-theory", "central-binomial-coefficients", "digit-representations", "erdos-problems" ], "branch": "number theory" } }, { "id": "leanpool:egrs75:Egrs75.MathlibAPI.irrational_log_div_log", "name": "Egrs75.MathlibAPI.irrational_log_div_log", "kind": "", "origin": "overlay:leanpool", "entry": "egrs75", "entry_name": "The Erdős–Graham–Ruzsa–Straus two-prime theorem", "repository": "https://github.com/lyfar/egrs75-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Egrs75", "lines": null, "statement": "", "doc": "A complete machine-checked proof that for any two distinct odd primes p and q there are infinitely many n with binom(2n,n) coprime to pq, including the iterated carry-controlled digit-clearing step that Bloom and Croot defer to the 1975 paper, via a per-move lexicographic measure, an elementary staircase lemma, and a Dirichlet-approximation seed.", "informal": "Proves that the ratio of logarithms of two distinct primes is irrational.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Egrs75", "classification": { "msc2020": [ "11A63", "11B65" ], "tags": [ "number-theory", "central-binomial-coefficients", "digit-representations", "erdos-problems" ], "branch": "number theory" } }, { "id": "leanpool:erdos-97-convex-octagon:Erdos97Octagon.erdos97_convex_octagon", "name": "Erdos97Octagon.erdos97_convex_octagon", "kind": "", "origin": "overlay:leanpool", "entry": "erdos-97-convex-octagon", "entry_name": "The Convex-Octagon Case of Erdős Problem 97", "repository": "https://github.com/lyfar/erdos-97-octagon-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos97ConvexOctagon", "lines": null, "statement": "", "doc": "Proves that among eight labelled points in the Euclidean plane in convex position, at least one point has no four other points at a common distance. The development reduces a geometric counterexample to a finite incidence model, classifies normalized row patterns up to symmetry, proves geometric and algebraic obstructions, and excludes seven canonical first-row masks with a kernel-audited exhaustive search over the remaining legal row masks and checked geometric obstruction witnesses. This is the modern four-equidistant formulation; Erdős's 1946 paper posed the stronger three-equidistant version. This is the convex-octagon case only; the general Erdős Problem 97 remains open.", "informal": "Among eight labelled points in the Euclidean plane in convex position, some point has no four other points at a common distance.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos97ConvexOctagon", "classification": { "msc2020": [ "51K05", "52A10" ], "tags": [ "discrete-geometry", "distance-geometry", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos-moser-distinct-subset-sums:LeanPool.ErdosMoser.leoMoserVarianceBound", "name": "LeanPool.ErdosMoser.leoMoserVarianceBound", "kind": "", "origin": "overlay:leanpool", "entry": "erdos-moser-distinct-subset-sums", "entry_name": "Leo Moser's finite distinct-subset-sums inequality", "repository": "https://github.com/lyfar/erdos-moser-distinct-subset-sums-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ErdosMoser", "lines": null, "statement": "", "doc": "Formalizes the exact finite sum-of-squares inequality that Guy's 1982 account states as Theorem 2 and attributes to Leo Moser: a finite set A of natural numbers with distinct subset sums satisfies 4^|A| - 1 <= 3 * sum(a^2). It then derives direct and square-root lower bounds for the largest element of A. This does not claim the stronger historical asymptotic bound attributed jointly to Erdős and Moser, and Erdős Problem 1 remains open.", "informal": "If a finite set A of natural numbers has distinct subset sums, then 4^|A| - 1 is at most three times the sum of the squares of the elements of A.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/ErdosMoser", "classification": { "msc2020": [ "11B13", "11B75" ], "tags": [ "additive-combinatorics", "number-theory", "distinct-subset-sums", "erdos-problems" ], "branch": "additive combinatorics" } }, { "id": "leanpool:erdos-moser-distinct-subset-sums:LeanPool.ErdosMoser.erdosMoserMaxBound", "name": "LeanPool.ErdosMoser.erdosMoserMaxBound", "kind": "", "origin": "overlay:leanpool", "entry": "erdos-moser-distinct-subset-sums", "entry_name": "Leo Moser's finite distinct-subset-sums inequality", "repository": "https://github.com/lyfar/erdos-moser-distinct-subset-sums-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ErdosMoser", "lines": null, "statement": "", "doc": "Formalizes the exact finite sum-of-squares inequality that Guy's 1982 account states as Theorem 2 and attributes to Leo Moser: a finite set A of natural numbers with distinct subset sums satisfies 4^|A| - 1 <= 3 * sum(a^2). It then derives direct and square-root lower bounds for the largest element of A. This does not claim the stronger historical asymptotic bound attributed jointly to Erdős and Moser, and Erdős Problem 1 remains open.", "informal": "For a nonempty finite set A of natural numbers with distinct subset sums, 4^|A| - 1 is at most 3|A| times the square of its largest element.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/ErdosMoser", "classification": { "msc2020": [ "11B13", "11B75" ], "tags": [ "additive-combinatorics", "number-theory", "distinct-subset-sums", "erdos-problems" ], "branch": "additive combinatorics" } }, { "id": "leanpool:erdos-moser-distinct-subset-sums:LeanPool.ErdosMoser.erdosMoserMaxSqrt", "name": "LeanPool.ErdosMoser.erdosMoserMaxSqrt", "kind": "", "origin": "overlay:leanpool", "entry": "erdos-moser-distinct-subset-sums", "entry_name": "Leo Moser's finite distinct-subset-sums inequality", "repository": "https://github.com/lyfar/erdos-moser-distinct-subset-sums-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.ErdosMoser", "lines": null, "statement": "", "doc": "Formalizes the exact finite sum-of-squares inequality that Guy's 1982 account states as Theorem 2 and attributes to Leo Moser: a finite set A of natural numbers with distinct subset sums satisfies 4^|A| - 1 <= 3 * sum(a^2). It then derives direct and square-root lower bounds for the largest element of A. This does not claim the stronger historical asymptotic bound attributed jointly to Erdős and Moser, and Erdős Problem 1 remains open.", "informal": "The equivalent square-root lower bound on the largest element: sqrt((4^|A| - 1)/(3|A|)) is at most max(A).", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/ErdosMoser", "classification": { "msc2020": [ "11B13", "11B75" ], "tags": [ "additive-combinatorics", "number-theory", "distinct-subset-sums", "erdos-problems" ], "branch": "additive combinatorics" } }, { "id": "leanpool:erdos-tuza-valtr:ErdosTuzaValtr.main", "name": "ErdosTuzaValtr.main", "kind": "", "origin": "overlay:leanpool", "entry": "erdos-tuza-valtr", "entry_name": "The Erdős–Tuza–Valtr conjecture", "repository": 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"kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos1196", "classification": { "msc2020": [ "11N25", "05D05" ], "tags": [ "number-theory", "combinatorics", "analysis" ], "branch": "analytic number theory" } }, { "id": "leanpool:erdos132-convex-k3:LeanPool.Erdos132ConvexK3.convex_top_three_degree_six", "name": "LeanPool.Erdos132ConvexK3.convex_top_three_degree_six", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-convex-k3", "entry_name": "Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures", "repository": "https://github.com/lyfar/erdos-132-convex-k3-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132ConvexK3", "lines": null, "statement": "", "doc": "Proves that every finite cyclically strictly convex planar configuration has a vertex incident to at most six edges from its three 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It does not settle the general-k suggestion or the general Erdős problem 132.", "informal": "For every nonempty finite cyclically strictly convex planar configuration with three strictly ordered largest squared-distance classes d₁, d₂, and d₃, some vertex is incident to at most six pairs whose squared distance is one of those three classes.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132ConvexK3", "classification": { "msc2020": [ "52C10", "05C12" ], "tags": [ "discrete-geometry", "distance-graphs", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-convex-k3:LeanPool.Erdos132ConvexK3.convex_top_three_draft_reduction_complete", "name": "LeanPool.Erdos132ConvexK3.convex_top_three_draft_reduction_complete", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-convex-k3", "entry_name": "Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures", "repository": "https://github.com/lyfar/erdos-132-convex-k3-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132ConvexK3", "lines": null, "statement": "", "doc": "Proves that every finite cyclically strictly convex planar configuration has a vertex incident to at most six edges from its three largest distance classes. 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It does not settle the general-k suggestion or the general Erdős problem 132.", "informal": "Cyclic strict convexity and the three-largest-distance-class hypothesis construct the full maximal-gap, thirteen-word reduction package together with route-specific closure evidence, without any remaining coordinated-majorant or geometric-adapter assumption.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132ConvexK3", "classification": { "msc2020": [ "52C10", "05C12" ], "tags": [ "discrete-geometry", "distance-graphs", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-convex-k3:LeanPool.Erdos132ConvexK3.global_thirteen_word_closure_complete", "name": "LeanPool.Erdos132ConvexK3.global_thirteen_word_closure_complete", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-convex-k3", "entry_name": "Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures", "repository": "https://github.com/lyfar/erdos-132-convex-k3-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132ConvexK3", "lines": null, "statement": "", "doc": "Proves that every finite cyclically strictly convex planar configuration has a vertex incident to at most six edges from its three largest distance classes. 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It does not settle the general-k suggestion or the general Erdős problem 132.", "informal": "Every one of the thirteen canonical maximal-gap cover words transports to its routed full-two-rung, anti-saturation, terminal-cage, or four-edge-cage realization, and hence supplies the corresponding degree bound.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132ConvexK3", "classification": { "msc2020": [ "52C10", "05C12" ], "tags": [ "discrete-geometry", "distance-graphs", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-convex-k3:LeanPool.Erdos132ConvexK3.concrete_word_closures", "name": "LeanPool.Erdos132ConvexK3.concrete_word_closures", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-convex-k3", "entry_name": "Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures", "repository": "https://github.com/lyfar/erdos-132-convex-k3-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132ConvexK3", "lines": null, "statement": "", "doc": "Proves that every finite cyclically strictly convex planar configuration has a vertex incident to at most six edges from its three largest distance classes. 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Each inhabited routed predicate supplies a vertex of degree at most six, with the one-penultimate route proving the stronger bound five.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132ConvexK3", "classification": { "msc2020": [ "52C10", "05C12" ], "tags": [ "discrete-geometry", "distance-graphs", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-convex-k3:LeanPool.Erdos132ConvexK3.fullTwoRung_realization_degree_le_six", "name": "LeanPool.Erdos132ConvexK3.fullTwoRung_realization_degree_le_six", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-convex-k3", "entry_name": "Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures", "repository": "https://github.com/lyfar/erdos-132-convex-k3-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132ConvexK3", "lines": null, "statement": "", "doc": "Proves that every finite cyclically strictly convex planar configuration has a vertex incident to at most six edges from its three largest distance classes. 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A rigid normalization derives the canonical coordinates, the diameter branch is impossible, the long branch is counted from four raw boundary arcs, and the short branch contradicts the exact two-rung identity and metric dichotomy.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132ConvexK3", "classification": { "msc2020": [ "52C10", "05C12" ], "tags": [ "discrete-geometry", "distance-graphs", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-convex-k3:LeanPool.Erdos132ConvexK3.convex_k3_degree_six_of_reduction", "name": "LeanPool.Erdos132ConvexK3.convex_k3_degree_six_of_reduction", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-convex-k3", "entry_name": "Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures", "repository": "https://github.com/lyfar/erdos-132-convex-k3-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132ConvexK3", "lines": null, "statement": "", "doc": "Proves that every finite cyclically strictly convex planar configuration has a vertex incident to at most six edges from its three largest distance classes. 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Its hypothesis is now discharged by convex_top_three_draft_reduction_complete, yielding the unconditional theorem convex_top_three_degree_six; ErLV89 itself proves only the degree-eight bound obtained from 3k−1 when k=3.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132ConvexK3", "classification": { "msc2020": [ "52C10", "05C12" ], "tags": [ "discrete-geometry", "distance-graphs", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-convex-k3:LeanPool.Erdos132ConvexK3.coordinated_exceptional_count_rigidity", "name": "LeanPool.Erdos132ConvexK3.coordinated_exceptional_count_rigidity", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-convex-k3", "entry_name": "Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures", "repository": "https://github.com/lyfar/erdos-132-convex-k3-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132ConvexK3", "lines": null, "statement": "", "doc": "Proves that every finite cyclically strictly convex planar configuration has a vertex incident to at most six edges from its three largest distance classes. 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It does not settle the general-k suggestion or the general Erdős problem 132.", "informal": "Assuming the three top distance classes and TopThreeAdjacent for both starting edges z–x and t–u, each forced two-move chain in an exceptional inner-move case has the rank ladder from the third-largest through the second-largest to the largest distance, while its proposed intermediate exchange edge remains nonterminal.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132ConvexK3", "classification": { "msc2020": [ "52C10", "05C12" ], "tags": [ "discrete-geometry", "distance-graphs", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-convex-k3:LeanPool.Erdos132ConvexK3.heptagon_nondiameter_edge_is_majorant", "name": "LeanPool.Erdos132ConvexK3.heptagon_nondiameter_edge_is_majorant", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-convex-k3", "entry_name": "Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures", "repository": "https://github.com/lyfar/erdos-132-convex-k3-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132ConvexK3", "lines": null, "statement": "", "doc": "Proves that every finite cyclically strictly convex planar configuration has a vertex incident to at most six edges from its three largest distance classes. 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It does not settle the general-k suggestion or the general Erdős problem 132.", "informal": "In an explicit rational convex heptagon, a second-largest-distance edge is already terminal under both permitted cover moves, refuting the stronger claim that every non-diameter top-three edge admits a longer cover.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132ConvexK3", "classification": { "msc2020": [ "52C10", "05C12" ], "tags": [ "discrete-geometry", "distance-graphs", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-convex-k3:LeanPool.Erdos132ConvexK3.Witnesses.ninePoint_top_three", "name": "LeanPool.Erdos132ConvexK3.Witnesses.ninePoint_top_three", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-convex-k3", "entry_name": "Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures", "repository": "https://github.com/lyfar/erdos-132-convex-k3-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132ConvexK3", "lines": null, "statement": "", "doc": "Proves that every finite cyclically strictly convex planar configuration has a vertex incident to at most six edges from its three largest distance classes. 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It does not settle the general-k suggestion or the general Erdős problem 132.", "informal": "An explicit rational nine-point regression configuration realizes squared-distance classes 855625, 838525, and 811165 as its three largest classes.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132ConvexK3", "classification": { "msc2020": [ "52C10", "05C12" ], "tags": [ "discrete-geometry", "distance-graphs", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-convex-k3:LeanPool.Erdos132ConvexK3.Witnesses.rationalHexagon_top_three", "name": "LeanPool.Erdos132ConvexK3.Witnesses.rationalHexagon_top_three", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-convex-k3", "entry_name": "Convex Three-Distance Degree-Six Theorem and Exceptional-Word Closures", "repository": "https://github.com/lyfar/erdos-132-convex-k3-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132ConvexK3", "lines": null, "statement": "", "doc": "Proves that every finite cyclically strictly convex planar configuration has a vertex incident to at most six edges from its three largest distance classes. The proof selects a maximal-gap vertex and coordinated majorants, closes all exceptional inner branches, performs the five-row enumeration, and transports all thirteen canonical cover words to four shared full-two-rung, anti-saturation, terminal-cage, and four-edge-cage predicates. It also proves exceptional-count rigidity and an explicit heptagon refutation of a tempting stronger majorant claim, with rational regression witnesses. This settles the all-n k=3 instance of the paper's suggested degree-2k convex bound; Morić--Pritchard proved the corresponding top-three edge bound for sufficiently large n. It does not settle the general-k suggestion or the general Erdős problem 132.", "informal": "An explicit rational six-point regression configuration realizes the three stated rational squared-distance values as its largest classes.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132ConvexK3", "classification": { "msc2020": [ "52C10", "05C12" ], "tags": [ "discrete-geometry", "distance-graphs", "erdos-problems", "convexity" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-n14:LeanPool.Erdos132N14.erdos132_for_fourteen_of_published_inputs", "name": "LeanPool.Erdos132N14.erdos132_for_fourteen_of_published_inputs", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-n14", "entry_name": "A Conditional Fourteen-Point Case of Erdős Problem 132", "repository": "https://github.com/lyfar/erdos-132-moment-obstruction-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132N14", "lines": null, "statement": "", "doc": "Formalizes the planar Hopf--Pannwitz diameter bound and proves the fourteen-point low-multiplicity conclusion conditional on exactly two remaining published interfaces: the known maximum-cardinality bound for planar sets with at most six distances and the Szöllősi--Östergård classification of thirteen-point planar six-distance sets. Internally derives the exact failure profile and endpoint deletion, constructs the three classified templates, eliminates the nonregular cases by multiplicity, and excludes the regular tridecagon extension by explicit second and fourth moments. The general Erdős problem remains open.", "informal": "Assuming precisely the published planar few-distance cardinality and Szöllősi--Östergård classification interfaces, every injectively labelled fourteen-point planar configuration realizes at least two distances whose unordered-pair multiplicities are at most fourteen.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132N14", "classification": { "msc2020": [ "52C10", "05D99" ], "tags": [ "discrete-geometry", "few-distance-sets", "erdos-problems", "planar-configurations" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos132-n14:LeanPool.Erdos132N14.hopfPannwitzLowMultiplicityDistance14", "name": "LeanPool.Erdos132N14.hopfPannwitzLowMultiplicityDistance14", "kind": "", "origin": "overlay:leanpool", "entry": "erdos132-n14", "entry_name": "A Conditional Fourteen-Point Case of Erdős Problem 132", "repository": "https://github.com/lyfar/erdos-132-moment-obstruction-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos132N14", "lines": null, "statement": "", "doc": "Formalizes the planar Hopf--Pannwitz diameter bound and proves the fourteen-point low-multiplicity conclusion conditional on exactly two remaining published interfaces: the known maximum-cardinality bound for planar sets with at most six distances and the Szöllősi--Östergård classification of thirteen-point planar six-distance sets. 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The general Erdős problem remains open.", "informal": "Every injectively labelled fourteen-point planar configuration has a realized distance represented by at most fourteen unordered pairs; the proof chooses that distance to be a diameter.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos132N14", "classification": { "msc2020": [ "52C10", "05D99" ], "tags": [ "discrete-geometry", "few-distance-sets", "erdos-problems", "planar-configurations" ], "branch": "discrete geometry" } }, { "id": "leanpool:erdos137:Erdos137.erdos137_finite", "name": "Erdos137.erdos137_finite", "kind": "", "origin": "overlay:leanpool", "entry": "erdos137", "entry_name": "Erdős Problem #137: powerful products of consecutive integers", "repository": "https://github.com/scottdhughes/erdos137", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos137", "lines": null, "statement": "", "doc": "Formalizes the abc-conditional finiteness results for Erdős Problem #137 (Erdős–Selfridge): a number N is powerful if every prime dividing N has its square dividing N, and the question is whether the product F(k,n) = n(n+1)⋯(n+k-1) of k ≥ 3 consecutive integers can be powerful for infinitely many n. 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"powerful-numbers", "abc-conjecture", "pell-equations" ], "branch": "number theory" } }, { "id": "leanpool:erdos403:Erdos403.erdos_403_sharp", "name": "Erdos403.erdos_403_sharp", "kind": "", "origin": "overlay:leanpool", "entry": "erdos403", "entry_name": "Sums of Distinct Factorials That Are Powers of Two (Erdos Problem 403)", "repository": "https://github.com/gotrevor/erdos-403", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos403", "lines": null, "statement": "", "doc": "Reconstructs the finiteness theorem for Erdos Problem 403: only finitely many sums of distinct factorials are powers of two, and every such power is at most 2^7 = 128 (the extremal solution 2! + 3! + 5! = 128). 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Derives a finite-dimensional inequality form of Farkas' lemma and proves packing attainment with equality to the covering infimum for finite covering and packing linear programs. 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The Farkas and LP statements overlap the pooled Duality development; the generic cone-closedness results provide the project's distinct reusable API.", "informal": "An infeasible finite system of nonnegative linear inequalities admits a nonnegative separating Farkas certificate.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos81PaperIContrib", "classification": { "msc2020": [ "52A20", "90C05" ], "tags": [ "convex-geometry", "linear-programming", "duality", "farkas-lemma" ], "branch": "convex geometry and optimization" } }, { "id": "leanpool:erdos81-paper-i-contrib:LeanPool.Erdos81PaperIContrib.covering_packing_duality", "name": "LeanPool.Erdos81PaperIContrib.covering_packing_duality", "kind": "", "origin": "overlay:leanpool", "entry": "erdos81-paper-i-contrib", "entry_name": "Finitely generated cones and finite LP duality", "repository": "https://github.com/jtraverso/erdos-81-chordal-clique-partitions", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos81PaperIContrib", "lines": null, "statement": "", "doc": "Proves that finitely generated cones in real normed spaces are closed, including a conic Carathéodory reduction and PointedCone API forms. Derives a finite-dimensional inequality form of Farkas' lemma and proves packing attainment with equality to the covering infimum for finite covering and packing linear programs. The Farkas and LP statements overlap the pooled Duality development; the generic cone-closedness results provide the project's distinct reusable API.", "informal": "Given a finite nonnegative matrix with a strictly positive entry in every column and nonnegative capacities, there is a feasible nonnegative packing whose unit-objective value is maximal and equals the infimum cost of nonnegative coverings with unit demand.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Erdos81PaperIContrib", "classification": { "msc2020": [ "52A20", "90C05" ], "tags": [ "convex-geometry", "linear-programming", "duality", "farkas-lemma" ], "branch": "convex geometry and optimization" } }, { "id": "leanpool:erdos865:Erdos865.erdos865_upper_bound", "name": "Erdos865.erdos865_upper_bound", "kind": "", "origin": "overlay:leanpool", "entry": "erdos865", "entry_name": "A sharp 5/8 bound for Erdős Problem 865", "repository": "https://github.com/mrricky22/erdos-865-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Erdos865", "lines": null, "statement": "", "doc": "Formalizes the resolution of Erdős Problem 865 on pairwise sums. For A ⊆ {1,…,N}, a pairwise-sum triple is a set of distinct a, b, c ∈ A with a+b, a+c, b+c all in A, and A is triple-free when it has none. The development proves that every triple-free A ⊆ [1,N] satisfies 8·|A| ≤ 5·N + 53, that any A exceeding this bound contains a triple, and — via the construction A = [M,2M] ∪ [4M,8M] for N = 8M — that the constant 5/8 is optimal, so f₃(N) = 5N/8 + O(1). 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For A ⊆ {1,…,N}, a pairwise-sum triple is a set of distinct a, b, c ∈ A with a+b, a+c, b+c all in A, and A is triple-free when it has none. The development proves that every triple-free A ⊆ [1,N] satisfies 8·|A| ≤ 5·N + 53, that any A exceeding this bound contains a triple, and — via the construction A = [M,2M] ∪ [4M,8M] for N = 8M — that the constant 5/8 is optimal, so f₃(N) = 5N/8 + O(1). 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estimate.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/HSDInteriorPointLP", "classification": { "msc2020": [ "90C05", "90C51" ], "tags": [ "linear-programming", "interior-point-methods", "optimization", "homogeneous-self-dual" ], "branch": "linear optimization" } }, { "id": "leanpool:incompleteness:LeanPool.Incompleteness.goedelFirst", "name": "LeanPool.Incompleteness.goedelFirst", "kind": "", "origin": "overlay:leanpool", "entry": "incompleteness", "entry_name": "Gödel's First and Second Incompleteness Theorems", "repository": "https://github.com/FormalizedFormalLogic/Incompleteness", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Incompleteness", "lines": null, "statement": "", "doc": "Formalizes 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"entry_name": "Gödel's First and Second Incompleteness Theorems", "repository": "https://github.com/FormalizedFormalLogic/Incompleteness", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Incompleteness", "lines": null, "statement": "", "doc": "Formalizes Gödel's first and second incompleteness theorems for first-order arithmetic, building on formalized arithmetization and provability-logic infrastructure.", "informal": "Proves Gödel's second incompleteness theorem: such a theory cannot prove its own consistency.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Incompleteness", "classification": { "msc2020": [ "03F40", "03F30" ], "tags": [ "incompleteness", "provability", "first-order-arithmetic", 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"human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Isoperimetric", "classification": { "msc2020": [ "28A75", "52A40", "49Q20" ], "tags": [ "measure-theory", "geometric-inequalities" ], "branch": "geometric analysis" } }, { "id": "leanpool:istranscendentalpi:IsTranscendentalPi", "name": "IsTranscendentalPi", "kind": "", "origin": "overlay:leanpool", "entry": "istranscendentalpi", "entry_name": "The transcendence of π", "repository": "https://github.com/samuelborza/IsTranscendentalPi", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.IsTranscendentalPi", "lines": null, "statement": "", "doc": "Formalizes a self-contained proof, following Niven's 1939 argument, that the real and complex number π is transcendental over the rationals.", "informal": "Proves that the complex number π is transcendental over ℚ.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/IsTranscendentalPi", "classification": { "msc2020": [ "11A41" ], "tags": [ "transcendence", "pi", "number-theory", "niven" ], "branch": "number theory" } }, { "id": "leanpool:istranscendentalpi:IsTranscendentalPiReal", "name": "IsTranscendentalPiReal", "kind": "", "origin": "overlay:leanpool", "entry": "istranscendentalpi", "entry_name": "The transcendence of π", "repository": "https://github.com/samuelborza/IsTranscendentalPi", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.IsTranscendentalPi", "lines": null, "statement": "", "doc": "Formalizes a self-contained proof, following Niven's 1939 argument, that the real and complex number π is transcendental over the rationals.", "informal": "Proves 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"impl_module": "LeanPool.IsTranscendentalPi", "lines": null, "statement": "", "doc": "Formalizes a self-contained proof, following Niven's 1939 argument, that the real and complex number π is transcendental over the rationals.", "informal": "Produces a monic rational polynomial vanishing at iπ when iπ is algebraic.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/IsTranscendentalPi", "classification": { "msc2020": [ "11A41" ], "tags": [ "transcendence", "pi", "number-theory", "niven" ], "branch": "number theory" } }, { "id": "leanpool:jacobian-diffgeo:genus_eq_zero_iff_homeo", "name": "genus_eq_zero_iff_homeo", "kind": "", "origin": "overlay:leanpool", "entry": "jacobian-diffgeo", "entry_name": "The Jacobian of a Compact 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], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/JohnsonLindenstraussLean", "classification": { "msc2020": [ "68W20", "60G15", "68P05" ], "tags": [ "dimensionality-reduction", "random-projection", "johnson-lindenstrauss", "gaussian", "quantization" ], "branch": "probability" } }, { "id": "leanpool:johnson-lindenstrauss-lean:JL.johnson_lindenstrauss_pointset", "name": "JL.johnson_lindenstrauss_pointset", "kind": "", "origin": "overlay:leanpool", "entry": "johnson-lindenstrauss-lean", "entry_name": "The Johnson–Lindenstrauss Lemma and Quantized JL", "repository": "https://github.com/claytomode/johnson-lindenstrauss-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.JohnsonLindenstraussLean", "lines": null, "statement": "", "doc": "Formalizes the Johnson–Lindenstrauss lemma end to end — a Gaussian random projection preserves pairwise 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"informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/JohnsonLindenstraussLean", "classification": { "msc2020": [ "68W20", "60G15", "68P05" ], "tags": [ "dimensionality-reduction", "random-projection", "johnson-lindenstrauss", "gaussian", "quantization" ], "branch": "probability" } }, { "id": "leanpool:johnson-lindenstrauss-lean:JL.qjlEstimator_concentration_exp", "name": "JL.qjlEstimator_concentration_exp", "kind": "", "origin": "overlay:leanpool", "entry": "johnson-lindenstrauss-lean", "entry_name": "The Johnson–Lindenstrauss Lemma and Quantized JL", "repository": "https://github.com/claytomode/johnson-lindenstrauss-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.JohnsonLindenstraussLean", "lines": null, "statement": "", "doc": "Formalizes the Johnson–Lindenstrauss lemma end to end — a Gaussian random projection preserves pairwise distances of a finite point set up to a (1 ± ε) factor, so a distortion-ε embedding of any m-point set exists once k is logarithmic in m — together with the quantized JL (QJL) one-bit sign sketch: unbiasedness of the asymmetric 1-bit estimator and its sub-Gaussian distortion bound.", "informal": "Sub-Gaussian distortion bound for the QJL 1-bit estimator: deviation at least ε has probability at most 2·exp(-m·ε²/(π·‖q‖²)).", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/JohnsonLindenstraussLean", "classification": { "msc2020": [ "68W20", "60G15", "68P05" ], "tags": [ "dimensionality-reduction", "random-projection", "johnson-lindenstrauss", "gaussian", "quantization" ], "branch": "probability" } }, { "id": "leanpool:kalton-roberts:KaltonRoberts.KR_constant_lt", "name": "KaltonRoberts.KR_constant_lt", "kind": "", "origin": "overlay:leanpool", "entry": "kalton-roberts", "entry_name": "Halving the Kalton-Roberts upper bound", "repository": "https://github.com/boonsuan/KaltonRoberts", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.KaltonRoberts", "lines": null, "statement": "", "doc": "Formalizes the paper-style Boolean-algebra theorem that every approximately additive real-valued set function is uniformly close to a finitely additive signed measure, proving the certified Kalton-Roberts constant bound below 19.838.", "informal": "Proves the headline certified bound that the Kalton-Roberts constant is strictly less than 9919/500, i.e. 19.838.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/KaltonRoberts", "classification": { "msc2020": [ "46B20", "28A12", "05C35" ], "tags": [ "functional-analysis", "finitely-additive-measures", "kalton-roberts" ], "branch": "analysis" } }, { "id": "leanpool:kalton-roberts:KaltonRoberts.KR_constant_le_C₂", "name": "KaltonRoberts.KR_constant_le_C₂", "kind": "", "origin": "overlay:leanpool", "entry": "kalton-roberts", "entry_name": "Halving the Kalton-Roberts upper bound", "repository": "https://github.com/boonsuan/KaltonRoberts", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.KaltonRoberts", "lines": null, "statement": "", "doc": "Formalizes the paper-style Boolean-algebra theorem that every approximately additive real-valued set function is uniformly close to a finitely additive signed measure, proving the certified Kalton-Roberts constant bound below 19.838.", "informal": "Proves the exact rational upper bound C₂ = 694198146664396294486127753 / 34994834677886019996000000 for the Kalton-Roberts constant.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/KaltonRoberts", "classification": { "msc2020": [ "46B20", "28A12", "05C35" ], "tags": [ "functional-analysis", "finitely-additive-measures", "kalton-roberts" ], "branch": "analysis" } }, { "id": "leanpool:kalton-roberts:KaltonRoberts.set_algebra_bound_C₂", "name": "KaltonRoberts.set_algebra_bound_C₂", "kind": "", "origin": "overlay:leanpool", "entry": "kalton-roberts", "entry_name": "Halving the Kalton-Roberts upper bound", "repository": "https://github.com/boonsuan/KaltonRoberts", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.KaltonRoberts", "lines": null, "statement": "", "doc": "Formalizes the paper-style Boolean-algebra theorem that every approximately additive real-valued set function is uniformly close to a finitely additive signed measure, proving the certified Kalton-Roberts constant bound below 19.838.", "informal": "States the paper-style set-algebra formulation, where every approximately additive function on a Boolean subalgebra of sets is within C₂ times the error of a finitely additive signed measure.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/KaltonRoberts", "classification": { "msc2020": [ "46B20", "28A12", "05C35" ], "tags": [ "functional-analysis", "finitely-additive-measures", "kalton-roberts" ], "branch": "analysis" } }, { "id": "leanpool:krafftsieve:KrafftSieve.mu_min_lt_one_implies_tpc", "name": "KrafftSieve.mu_min_lt_one_implies_tpc", "kind": "", "origin": "overlay:leanpool", "entry": "krafftsieve", "entry_name": "A Conditional Sieve Criterion for Twin Primes via Krafft Geometry", "repository": "https://github.com/ElNando888/KrafftSieve", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.KrafftSieve", "lines": null, "statement": "", "doc": "Introduces an original conditional sieve criterion for the Twin Prime Conjecture. Using a centered modular alignment (termed the Krafft alignment), proves that if the Rayleigh-Ritz quotient over the orthogonal unit sphere satisfies muMin < 1, then a twin prime pair must exist in the corresponding interval.", "informal": "If there are infinitely many intervals where the optimal multidimensional weight achieves a ratio strictly less than 1, then there are infinitely many twin primes. This is a conditional reduction; the hypothesis muMin n < 1 is not established here for any n.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/KrafftSieve", "classification": { "msc2020": [ "11N05", "11N35" ], "tags": [ "analytic-number-theory", "sieve-theory", "twin-primes", "optimization" ], "branch": "analytic number theory" } }, { "id": "leanpool:krafftsieve:KrafftSieve.krafft_sieve_guarantee_with_mu_min", "name": "KrafftSieve.krafft_sieve_guarantee_with_mu_min", "kind": "", "origin": "overlay:leanpool", "entry": "krafftsieve", "entry_name": "A Conditional Sieve Criterion for Twin Primes via Krafft Geometry", "repository": "https://github.com/ElNando888/KrafftSieve", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.KrafftSieve", "lines": null, "statement": "", "doc": "Introduces an original conditional sieve criterion for the Twin Prime Conjecture. 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"entry": "kuramoto", "entry_name": "Finite-N Kuramoto Synchronization", "repository": "https://github.com/velvetmonkey/kuramoto-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Kuramoto", "lines": null, "statement": "", "doc": "Formalizes the finite-N Kuramoto model of coupled phase oscillators: the order parameter and its unit bound, the potential/force gradient identities, smoothness and local Lipschitz continuity of the vector field with ODE existence and uniqueness, Lyapunov descent along trajectories, pairwise and semicircle contraction estimates, Hebbian phase/weight dynamics, and full convergence to synchrony under all-to-all coupling from initial phases in an open semicircle.", "informal": "The Kuramoto order parameter has norm at most one for any phase configuration.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Kuramoto", "classification": { "msc2020": [ "34D06" ], "tags": [ "dynamical-systems", "synchronization", "kuramoto" ], "branch": "dynamical systems" } }, { "id": "leanpool:kuramoto:weighted_lyapunov_descent", "name": "weighted_lyapunov_descent", "kind": "", "origin": "overlay:leanpool", "entry": "kuramoto", "entry_name": "Finite-N Kuramoto Synchronization", "repository": "https://github.com/velvetmonkey/kuramoto-lean", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Kuramoto", "lines": null, "statement": "", "doc": "Formalizes the finite-N Kuramoto model of coupled phase oscillators: the order parameter and its unit bound, the potential/force gradient identities, smoothness and local Lipschitz continuity of the vector field with ODE existence and uniqueness, Lyapunov descent along trajectories, pairwise and semicircle contraction estimates, Hebbian phase/weight dynamics, and full convergence to synchrony under all-to-all coupling from initial phases in an open semicircle.", "informal": "The weighted Kuramoto potential is non-increasing along the gradient flow, since the sum of force times potential-derivative over all oscillators is at most zero.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Kuramoto", "classification": { "msc2020": [ "34D06" ], "tags": [ "dynamical-systems", "synchronization", "kuramoto" ], "branch": "dynamical systems" } }, { "id": "leanpool:kuramoto:allToAll_convergence_to_synchrony", "name": "allToAll_convergence_to_synchrony", "kind": "", "origin": "overlay:leanpool", "entry": "kuramoto", "entry_name": "Finite-N Kuramoto 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"human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Kuramoto", "classification": { "msc2020": [ "34D06" ], "tags": [ "dynamical-systems", "synchronization", "kuramoto" ], "branch": "dynamical systems" } }, { "id": "leanpool:lattice-triangle:analyticEngine_lower_bound", "name": "analyticEngine_lower_bound", "kind": "", "origin": "overlay:leanpool", "entry": "lattice-triangle", "entry_name": "On the paucity of lattice triangles", "repository": "https://github.com/AxiomMath/lattice-triangle", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LatticeTriangle", "lines": null, "statement": "", "doc": "Formalizes an analytic lower bound used in the paucity theorem for lattice triangles.", "informal": "Proves the analytic-engine lower bound underlying the lattice-triangle result.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LatticeTriangle", "classification": { "msc2020": [ "11H06", "11B30" ], "tags": [ "number-theory", "combinatorics" ], "branch": "additive combinatorics" } }, { "id": "leanpool:lean-booleanfun:LeanPool.LeanBooleanfun.BooleanFun.BV.dictator_of_condorcet_and_unanimous", "name": "LeanPool.LeanBooleanfun.BooleanFun.BV.dictator_of_condorcet_and_unanimous", "kind": "", "origin": "overlay:leanpool", "entry": "lean-booleanfun", "entry_name": "Analysis of Boolean functions in Lean", "repository": "https://github.com/roos-j/lean-booleanfun", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LeanBooleanfun", "lines": null, "statement": "", "doc": "Walsh-Fourier analysis on Boolean functions, including Arrow's theorem and BLR linearity testing.", "informal": "Arrow's theorem for two-candidate Boolean voting rules.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanBooleanfun", "classification": { "msc2020": [ "06E30", "42C10", "91B14" ], "tags": [ "boolean-functions", "fourier-analysis", "social-choice" ], "branch": "combinatorics" } }, { "id": "leanpool:lean-model-checking:LeanModelChecking.for_any_LTL_formula_exists_an_equivalent_NBW", "name": "LeanModelChecking.for_any_LTL_formula_exists_an_equivalent_NBW", "kind": "", "origin": "overlay:leanpool", "entry": "lean-model-checking", "entry_name": "Formalizations of theorems related to model checking", "repository": "https://github.com/kuruczgy/lean-model-checking", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LeanModelChecking", "lines": null, "statement": "", "doc": "Proves that every linear temporal logic formula has an equivalent finite-state nondeterministic Büchi automaton (via negation normal form, alternating Büchi automata, and the Miyano-Hayashi breakpoint construction), and that every linear-time property decomposes as the intersection of a safety property and a liveness property.", "informal": "For any linear temporal logic formula there exists an equivalent finite-state nondeterministic Büchi automaton, i.e. one whose state type is finite, accepting the same language.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanModelChecking", "classification": { "msc2020": [ "68Q60", "03B44", 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propositions.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanModelChecking", "classification": { "msc2020": [ "68Q60", "03B44", "68Q45" ], "tags": [ "model-checking", "linear-temporal-logic", "buchi-automaton", "safety-liveness", "omega-automata" ], "branch": "model checking" } }, { "id": "leanpool:lean-model-checking:LeanModelChecking.NBW", "name": "LeanModelChecking.NBW", "kind": "", "origin": "overlay:leanpool", "entry": "lean-model-checking", "entry_name": "Formalizations of theorems related to model checking", "repository": "https://github.com/kuruczgy/lean-model-checking", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LeanModelChecking", "lines": null, "statement": "", "doc": "Proves that every linear temporal logic formula has an equivalent finite-state nondeterministic Büchi automaton (via negation normal form, alternating Büchi automata, and the Miyano-Hayashi breakpoint construction), and that every linear-time property decomposes as the intersection of a safety property and a liveness property.", "informal": "A nondeterministic Büchi automaton over a given alphabet of letters.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanModelChecking", "classification": { "msc2020": [ "68Q60", "03B44", "68Q45" ], "tags": [ "model-checking", "linear-temporal-logic", "buchi-automaton", "safety-liveness", "omega-automata" ], "branch": "model checking" } }, { "id": 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"leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanModelChecking", "classification": { "msc2020": [ "68Q60", "03B44", "68Q45" ], "tags": [ "model-checking", "linear-temporal-logic", "buchi-automaton", "safety-liveness", "omega-automata" ], "branch": "model checking" } }, { "id": "leanpool:lean-model-checking:SafetyLivenessDecomposition.safety_liveness_decomposition", "name": "SafetyLivenessDecomposition.safety_liveness_decomposition", "kind": "", "origin": "overlay:leanpool", "entry": "lean-model-checking", "entry_name": "Formalizations of theorems related to model checking", "repository": "https://github.com/kuruczgy/lean-model-checking", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LeanModelChecking", "lines": null, "statement": "", "doc": "Proves that every linear temporal logic formula has an equivalent finite-state nondeterministic Büchi automaton (via negation normal form, alternating Büchi automata, and the Miyano-Hayashi breakpoint construction), and that every linear-time property decomposes as the intersection of a safety property and a liveness property.", "informal": "Every linear-time property is the intersection of a safety property and a liveness property.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanModelChecking", "classification": { "msc2020": [ "68Q60", "03B44", "68Q45" ], "tags": [ "model-checking", "linear-temporal-logic", "buchi-automaton", "safety-liveness", "omega-automata" ], "branch": "model checking" } }, { "id": "leanpool:lean-poly-abc:LeanPolyABC.Polynomial.abc", "name": "LeanPolyABC.Polynomial.abc", "kind": "", "origin": "overlay:leanpool", "entry": "lean-poly-abc", "entry_name": "Polynomial ABC (Mason–Stothers) and its corollaries", "repository": "https://github.com/seewoo5/lean-poly-abc", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LeanPolyABC", "lines": null, "statement": "", "doc": "Formalizes the Mason–Stothers theorem (the polynomial ABC theorem) and derives the polynomial Fermat's Last Theorem, the Fermat–Catalan inequality, Davenport's theorem, and a no-parametrization corollary.", "informal": "Mason–Stothers — for coprime non-constant polynomials with a+b=c, the max degree is below the degree of the radical of abc.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], 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"https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanComplexAnalysis", "classification": { "msc2020": [ "30A99", "31A05", "30C55" ], "tags": [ "complex-analysis", "harmonic-functions", "poisson-integral", "univalent-functions" ], "branch": "complex analysis" } }, { "id": "leanpool:leancomplexanalysis:LeanPool.LeanComplexAnalysis.HerglotzRiesz_representation_harmonic", "name": "LeanPool.LeanComplexAnalysis.HerglotzRiesz_representation_harmonic", "kind": "", "origin": "overlay:leanpool", "entry": "leancomplexanalysis", "entry_name": "Formalized Complex Analysis in Lean", "repository": "https://github.com/seb488/LeanComplexAnalysis", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LeanComplexAnalysis", "lines": null, "statement": "", "doc": "Formalizes Poisson integral formulas on the unit disc and arbitrary centered discs, Herglotz–Riesz representation, and Harnack's inequality on the unit disc, plus infrastructure for univalent function classes S and Σ.", "informal": "Proves that every positive harmonic function on the unit disc with value 1 at 0 has a unique Herglotz–Riesz representation by a probability measure on the unit circle.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanComplexAnalysis", "classification": { "msc2020": [ "30A99", "31A05", "30C55" ], "tags": [ "complex-analysis", "harmonic-functions", "poisson-integral", "univalent-functions" ], "branch": "complex analysis" } }, { "id": "leanpool:leanmodularforms:generalizedResidueTheorem", "name": "generalizedResidueTheorem", "kind": "", "origin": "overlay:leanpool", "entry": "leanmodularforms", "entry_name": "Modular forms and the generalized residue theorem", "repository": "https://github.com/CBirkbeck/LeanModularForms", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LeanModularForms", "lines": null, "statement": "", "doc": "Formalizes the Hungerbühler-Wasem generalized residue theorem for Cauchy principal value integrals through poles and applies it to the valence formula for modular forms on the modular group.", "informal": "Proves the Hungerbühler-Wasem generalized residue theorem for piecewise-C1 curves through poles.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanModularForms", "classification": { "msc2020": [ "11F11", "30E20" ], "tags": [ "modular-forms", "complex-analysis", "residue-theorem", "valence-formula" ], "branch": "complex analysis" } }, { "id": "leanpool:leanmodularforms:valence_formula_textbook_orbit_finsum", "name": "valence_formula_textbook_orbit_finsum", "kind": "", "origin": "overlay:leanpool", "entry": "leanmodularforms", "entry_name": "Modular forms and the generalized residue theorem", "repository": "https://github.com/CBirkbeck/LeanModularForms", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LeanModularForms", "lines": null, "statement": "", "doc": "Formalizes the Hungerbühler-Wasem generalized residue theorem for Cauchy principal value integrals through poles and applies it to the valence formula for modular forms on the modular group.", "informal": "Proves the valence formula for modular forms on the modular group.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanModularForms", "classification": { "msc2020": [ "11F11", "30E20" ], "tags": [ "modular-forms", "complex-analysis", "residue-theorem", "valence-formula" ], "branch": "complex analysis" } }, { "id": "leanpool:leanmodularforms:generalizedResidueTheorem_simplePoles", "name": "generalizedResidueTheorem_simplePoles", "kind": "", "origin": "overlay:leanpool", "entry": "leanmodularforms", "entry_name": "Modular forms and the generalized residue theorem", "repository": "https://github.com/CBirkbeck/LeanModularForms", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LeanModularForms", "lines": null, "statement": "", "doc": "Formalizes the Hungerbühler-Wasem generalized residue theorem for Cauchy principal value integrals through poles and applies it to the valence formula for modular forms on the modular group.", "informal": "Specializes the generalized residue theorem to the simple-pole case.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LeanModularForms", "classification": { "msc2020": [ "11F11", "30E20" ], "tags": [ "modular-forms", "complex-analysis", "residue-theorem", "valence-formula" ], "branch": "complex analysis" } }, { "id": "leanpool:lehmer-e10:LehmerE10.main_theorem", "name": "LehmerE10.main_theorem", "kind": "", "origin": "overlay:leanpool", "entry": "lehmer-e10", "entry_name": "Lehmer's Polynomial and the E10 Coxeter Element", "repository": "https://github.com/dillon-11/lehmer-E10", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LehmerE10", "lines": null, "statement": "", "doc": "Proves that Lehmer's degree-10 polynomial — the integer polynomial of smallest known Mahler measure above 1 — is irreducible over the integers, and that it is the characteristic polynomial of a Coxeter element of the E10 Weyl group (the rank-10 instance of McMullen). Against Mathlib's Mahler-measure API it computes M(L) exactly as the Salem root with the unconditional bounds 1 < M(L) < 7/5, states Lehmer's conjecture as a defined Prop (never assumed) with the strong form implying the gap form, and contrasts the E8 Coxeter element (order exactly 30) with the E10 element (infinite order in GL(10,Z)): the E-series crosses from torsion to a Salem matrix at rank 10, and the crossing value is Lehmer's number.", "informal": "Lehmer's polynomial L(x) = x^10 + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1 is irreducible over the integers, and the characteristic polynomial of a Coxeter element of the E10 Weyl group equals L.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LehmerE10", "classification": { "msc2020": [ "11R06", "11C08", "20F55" ], "tags": [ "number-theory", "mahler-measure", "salem-numbers", "coxeter-groups" ], "branch": "number theory" } }, { "id": "leanpool:lehmer-e10:LehmerE10.lehmer_mahlerMeasure", "name": "LehmerE10.lehmer_mahlerMeasure", "kind": "", "origin": "overlay:leanpool", "entry": "lehmer-e10", "entry_name": "Lehmer's Polynomial and the E10 Coxeter Element", "repository": "https://github.com/dillon-11/lehmer-E10", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LehmerE10", "lines": null, "statement": "", "doc": "Proves that Lehmer's degree-10 polynomial — the integer polynomial of smallest known Mahler measure above 1 — is irreducible over the integers, and that it is the characteristic polynomial of a Coxeter element of the E10 Weyl group (the rank-10 instance of McMullen). Against Mathlib's Mahler-measure API it computes M(L) exactly as the Salem root with the unconditional bounds 1 < M(L) < 7/5, states Lehmer's conjecture as a defined Prop (never assumed) with the strong form implying the gap form, and contrasts the E8 Coxeter element (order exactly 30) with the E10 element (infinite order in GL(10,Z)): the E-series crosses from torsion to a Salem matrix at rank 10, and the crossing value is Lehmer's number.", "informal": "The Mahler measure of L equals its Salem root mu, with the unconditional bounds 1 < M(L) < 7/5.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LehmerE10", "classification": { "msc2020": [ "11R06", "11C08", "20F55" ], "tags": [ "number-theory", "mahler-measure", "salem-numbers", "coxeter-groups" ], "branch": "number theory" } }, { "id": "leanpool:lentil:TLA.always", "name": "TLA.always", "kind": "", "origin": "overlay:leanpool", "entry": "lentil", "entry_name": "Lentil: Temporal Logic of Actions in Lean 4", "repository": "https://github.com/verse-lab/Lentil", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Lentil", "lines": null, "statement": "", "doc": "A shallow embedding of Leslie Lamport's Temporal Logic of Actions (TLA) in Lean 4, with the temporal modalities, standard propositional and temporal proof rules, weak-fairness and leads-to reasoning, and an interactive proof mode for TLA judgments.", "informal": "Defines the TLA \"always\" (box) temporal modality, asserting a predicate holds on every suffix of an execution.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Lentil", "classification": { "msc2020": [ "03B44", "68Q60" ], "tags": [ "temporal-logic", "tla", "formal-verification", "proof-mode" ], "branch": "temporal logic" } }, { "id": "leanpool:lentil:TLA.eventually", "name": "TLA.eventually", "kind": "", "origin": "overlay:leanpool", "entry": "lentil", "entry_name": "Lentil: Temporal Logic of Actions in Lean 4", "repository": "https://github.com/verse-lab/Lentil", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Lentil", "lines": null, "statement": "", "doc": "A shallow embedding of Leslie Lamport's Temporal Logic of Actions (TLA) in Lean 4, with the temporal modalities, standard propositional and temporal proof rules, weak-fairness and leads-to reasoning, and an interactive proof mode for TLA judgments.", "informal": "Defines the TLA \"eventually\" (diamond) temporal modality, asserting a predicate holds on some suffix of an execution.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Lentil", "classification": { "msc2020": [ "03B44", "68Q60" ], "tags": [ "temporal-logic", "tla", "formal-verification", "proof-mode" ], "branch": "temporal logic" } }, { "id": "leanpool:lentil:TLA.always_induction", "name": "TLA.always_induction", "kind": "", "origin": "overlay:leanpool", "entry": "lentil", "entry_name": "Lentil: Temporal Logic of Actions in Lean 4", "repository": "https://github.com/verse-lab/Lentil", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Lentil", "lines": null, "statement": "", "doc": "A shallow embedding of Leslie Lamport's Temporal Logic of Actions (TLA) in Lean 4, with the temporal modalities, 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"impl_module": "LeanPool.Lentil", "lines": null, "statement": "", "doc": "A shallow embedding of Leslie Lamport's Temporal Logic of Actions (TLA) in Lean 4, with the temporal modalities, standard propositional and temporal proof rules, weak-fairness and leads-to reasoning, and an interactive proof mode for TLA judgments.", "informal": "The WF1 weak-fairness proof rule deriving a leads-to property from a next-state action under weak fairness.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Lentil", "classification": { "msc2020": [ "03B44", "68Q60" ], "tags": [ "temporal-logic", "tla", "formal-verification", "proof-mode" ], "branch": "temporal logic" } }, { "id": "leanpool:lentil:TLA.wf_as_leads_to", "name": "TLA.wf_as_leads_to", "kind": "", "origin": "overlay:leanpool", "entry": "lentil", "entry_name": "Lentil: Temporal Logic of Actions in Lean 4", "repository": "https://github.com/verse-lab/Lentil", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Lentil", "lines": null, "statement": "", "doc": "A shallow embedding of Leslie Lamport's Temporal Logic of Actions (TLA) in Lean 4, with the temporal modalities, standard propositional and temporal proof rules, weak-fairness and leads-to reasoning, and an interactive proof mode for TLA judgments.", "informal": "Characterizes weak fairness of an action as a leads-to property between its enabledness and its occurrence.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Lentil", "classification": { "msc2020": [ "03B44", "68Q60" ], "tags": [ "temporal-logic", "tla", "formal-verification", "proof-mode" ], "branch": "temporal logic" } }, { "id": "leanpool:lentil:TLA.leads_to_trans", "name": "TLA.leads_to_trans", "kind": "", "origin": "overlay:leanpool", "entry": "lentil", "entry_name": "Lentil: Temporal Logic of Actions in Lean 4", "repository": "https://github.com/verse-lab/Lentil", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Lentil", "lines": null, "statement": "", "doc": "A shallow embedding of Leslie Lamport's Temporal Logic of Actions (TLA) in Lean 4, with the temporal modalities, standard propositional and temporal proof rules, weak-fairness and leads-to reasoning, and an interactive proof mode for TLA judgments.", "informal": "Transitivity of the leads-to relation.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Lentil", "classification": { "msc2020": [ "03B44", "68Q60" ], "tags": [ "temporal-logic", "tla", "formal-verification", "proof-mode" ], "branch": "temporal logic" } }, { "id": "leanpool:lowdimsolvclassification:LieAlgebra.Dim3.classification", "name": "LieAlgebra.Dim3.classification", "kind": "", "origin": "overlay:leanpool", "entry": "lowdimsolvclassification", "entry_name": "Classification of low-dimensional solvable Lie algebras", "repository": "https://github.com/LieLean/LowDimSolvClassification", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.LowDimSolvClassification", "lines": null, "statement": "", "doc": "Formalises the classification of solvable Lie algebras up to dimension three.", "informal": "Classifies three-dimensional solvable Lie algebras over arbitrary fields.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/LowDimSolvClassification", "classification": { "msc2020": [ "17B30" ], "tags": [ "lie-algebras", "solvable", "classification" ], "branch": "lie theory" } }, { "id": "leanpool:main-theorem-of-polytopes:MainTheoremOfPolytopes", "name": "MainTheoremOfPolytopes", "kind": "", "origin": "overlay:leanpool", "entry": "main-theorem-of-polytopes", "entry_name": "The main theorem of polytopes", "repository": "https://github.com/Jun2M/Main-theorem-of-polytopes", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Polytopes", "lines": null, "statement": "", "doc": "Formalizes the Minkowski–Weyl theorem — the equivalence between V-polytopes (convex hulls of finite point sets) and bounded H-polytopes (finite intersections of half-spaces) in a finite-dimensional real vector space.", "informal": "The Minkowski–Weyl theorem — a set is a V-polytope iff it is a bounded H-polytope.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Polytopes", "classification": { "msc2020": [ "52B11", "52A20" ], "tags": [ "convex-geometry", "discrete-geometry", "polytopes" ], "branch": "convex geometry" } }, { "id": "leanpool:main-theorem-of-polytopes:Vpolytope_of_Hpolytope", "name": "Vpolytope_of_Hpolytope", "kind": "", "origin": "overlay:leanpool", "entry": "main-theorem-of-polytopes", "entry_name": "The main theorem of polytopes", "repository": "https://github.com/Jun2M/Main-theorem-of-polytopes", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.Polytopes", "lines": null, "statement": "", "doc": "Formalizes the Minkowski–Weyl theorem — the equivalence between V-polytopes (convex hulls of finite point sets) and bounded H-polytopes (finite intersections of half-spaces) in a finite-dimensional real vector space.", "informal": "Every bounded H-polytope (finite intersection of half-spaces) is the convex hull of a finite point set.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Polytopes", "classification": { "msc2020": [ "52B11", "52A20" ], "tags": [ "convex-geometry", "discrete-geometry", "polytopes" ], "branch": "convex geometry" } }, { "id": 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metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/MinModulusUniqueMultisetSum", "classification": { "msc2020": [ "11B75", "05D99" ], "tags": [ "additive-combinatorics", "multiset-sums", "finite-groups", "permanent" ], "branch": "additive combinatorics" } }, { "id": "leanpool:min-modulus-unique-multiset-sum:MinModulus.elementaryAbelianTwoGroups_optimal", "name": "MinModulus.elementaryAbelianTwoGroups_optimal", "kind": "", "origin": "overlay:leanpool", "entry": "min-modulus-unique-multiset-sum", "entry_name": "Minimum modulus for the unique multiset-sum problem", "repository": "https://github.com/jarfo/min-modulus", "commit": "", "lean": "", "spec_file": "", "impl_module": "LeanPool.MinModulusUniqueMultisetSum", "lines": null, "statement": "", "doc": 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"LeanPool.Monsky", "lines": null, "statement": "", "doc": "Formalizes Monsky's theorem that a square can be dissected into n triangles of equal area if and only if n is nonzero and even, via a 2-adic valuation, a three-coloring of the unit square, and a rainbow-triangle argument.", "informal": "Proves that the unit square admits a dissection into n equal-area triangles if and only if n is nonzero and even.", "informal_source": "leanpool metadata", "checks": { "upstream": "leanpool" }, "upstream_checks": { "comparator": false, "kernels": [ "lean" ], "human_signoff": false, "llm_review": true, "sorry_free": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://github.com/Vilin97/lean-pool/tree/main/LeanPool/Monsky", "classification": { "msc2020": [ "52C20", "05B45" ], "tags": [ "geometry", "combinatorics", "measure-theory" ], "branch": "combinatorial geometry" } }, { "id": "leanpool:moser-lattice-colorings:LeanPool.MoserLatticeColorings.MoserRing.both_colorings_are_geometric", 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Here representations are unordered two-element multisets from A, repeated summands are allowed, and \"no unique sum\" means that every such multiset has a distinct equal-sum mate.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-18-000003", "classification": { "arxiv": [ "math.CO", "math.NT" ], "msc2020": [ "11B30", "11B34" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000001:jordan_curve", "name": "jordan_curve", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000001", "entry_name": "rkirov/jordan_pick", "repository": "https://github.com/rkirov/jordan_pick", "commit": "b3c9b7cf7358bf81a077d78ad67e6e8247869ddd", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "The Jordan curve theorem, formalized in Lean 4 against Mathlib: a continuous injective map from the circle into the plane has a complement with exactly two connected components. The hypothesis is bare continuity and injectivity — no smoothness, piecewise-linearity or rectifiability — which is what makes the theorem hard and what separates it from the polygonal case. The proof follows Maehara's reduction to the Brouwer fixed point theorem, via a crossing lemma for transversal paths in a rectangle and the fact that every component has the curve as its boundary, together with a farthest-pair normalization that pins down exactly one bounded component. Because Mathlib has no Brouwer fixed point theorem at the pinned revision, Brouwer is built here from the ground up: covering-space path lifting for the circle giving π₁(S¹) ≅ ℤ, then no-retraction of the disk onto its boundary, then Brouwer on the disk by ray-retraction, then the general nonempty-compact-convex case by nearest-point projection. The statement is expressed in Mathlib's own vocabulary. The proof is sorry-free and depends on Lean's three standard axioms (propext, Classical.choice, Quot.sound) and nothing else.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000001", "classification": { "arxiv": [ "math.GT", "math.GN" ], "msc2020": [ "54F65", "54F15", "54H25", "55M20", "55M25", "54D05" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000002:erdos501_closed_infinite", "name": "erdos501_closed_infinite", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000002", "entry_name": "elliotglazer/erdos501", "repository": "https://github.com/elliotglazer/erdos501", "commit": "218d1c1e46f77d4db80e566d1721782e85b94a17", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Erdős problem #501 (Erdős 1961; Erdős–Hajnal 1971, Problem 38): for every real x let A_x be a bounded set of reals of Lebesgue outer measure < 1; must there be an infinite independent set, i.e. an infinite X ⊆ ℝ with x ∉ A_y for all distinct x, y ∈ X? And if the sets A_x are closed of measure < 1, must there be an independent set of size 3? This repository formalizes both answers. Second question: yes — closed sets of measure < 1 admit an infinite independent set (Newelski–Pawlikowski–Seredyński 1987; no boundedness is needed), hence one of size 3. First question: independent of ZFC. Under CH (as ℵ₁ = 𝔠) Hechler's 1972 construction gives a family of bounded sets of outer measure < 1 (in fact countable and null) with no infinite independent set — a Mathlib-level theorem — and the negation of the first question holds in the Boolean-valued model of the collapse algebra Col(ω₁, 𝒫(ω)); after adding 𝔠⁺ random reals (Glazer and Sol 2026, adapting the paper's ω₂ random reals over a CH ground) the first question has a positive answer, so it holds in the Boolean-valued model of the random algebra with 𝔠⁺ coordinates. Both Boolean-valued models are built in a Lean 4 port of Han–van Doorn's Flypitch (the framework of their independence-of-CH proof), and Flypitch's completeness theorem turns them into two-valued models. The comparator challenge states the results in Mathlib alone: the language of set theory, the theory ZFC (Flypitch's axiomatization: extensionality, empty set, ordered pairs, union, power set, infinity, regularity, Zorn's lemma, strong collection) and the sentence Erdos501 (\"every complete ordered field has the Erdős property\") are defined in Mathlib's ModelTheory, and independence is stated semantically as ¬ (ZFC ⊨ᵇ Erdos501) ∧ ¬ (ZFC ⊨ᵇ ∼Erdos501); a faithfulness theorem shows that in Mathlib's ZFSet the sentence is equivalent to the Mathlib statement of the first question, verbatim the proposition of google-deepmind/formal-conjectures. All targets are proved from propext, Classical.choice and Quot.sound only. As far as we know this is the first Erdős problem whose resolution is a formally verified independence result.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000002", "classification": { "arxiv": [ "math.LO", "math.CO" ], "msc2020": [ "03E35", "03E75", "03E40", "03E50", "05D10", "28A05" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000002:erdos501_closed_size3", "name": "erdos501_closed_size3", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000002", "entry_name": "elliotglazer/erdos501", "repository": "https://github.com/elliotglazer/erdos501", "commit": "218d1c1e46f77d4db80e566d1721782e85b94a17", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Erdős problem #501 (Erdős 1961; Erdős–Hajnal 1971, Problem 38): for every real x let A_x be a bounded set of reals of Lebesgue outer measure < 1; must there be an infinite independent set, i.e. an infinite X ⊆ ℝ with x ∉ A_y for all distinct x, y ∈ X? And if the sets A_x are closed of measure < 1, must there be an independent set of size 3? This repository formalizes both answers. Second question: yes — closed sets of measure < 1 admit an infinite independent set (Newelski–Pawlikowski–Seredyński 1987; no boundedness is needed), hence one of size 3. First question: independent of ZFC. Under CH (as ℵ₁ = 𝔠) Hechler's 1972 construction gives a family of bounded sets of outer measure < 1 (in fact countable and null) with no infinite independent set — a Mathlib-level theorem — and the negation of the first question holds in the Boolean-valued model of the collapse algebra Col(ω₁, 𝒫(ω)); after adding 𝔠⁺ random reals (Glazer and Sol 2026, adapting the paper's ω₂ random reals over a CH ground) the first question has a positive answer, so it holds in the Boolean-valued model of the random algebra with 𝔠⁺ coordinates. Both Boolean-valued models are built in a Lean 4 port of Han–van Doorn's Flypitch (the framework of their independence-of-CH proof), and Flypitch's completeness theorem turns them into two-valued models. The comparator challenge states the results in Mathlib alone: the language of set theory, the theory ZFC (Flypitch's axiomatization: extensionality, empty set, ordered pairs, union, power set, infinity, regularity, Zorn's lemma, strong collection) and the sentence Erdos501 (\"every complete ordered field has the Erdős property\") are defined in Mathlib's ModelTheory, and independence is stated semantically as ¬ (ZFC ⊨ᵇ Erdos501) ∧ ¬ (ZFC ⊨ᵇ ∼Erdos501); a faithfulness theorem shows that in Mathlib's ZFSet the sentence is equivalent to the Mathlib statement of the first question, verbatim the proposition of google-deepmind/formal-conjectures. All targets are proved from propext, Classical.choice and Quot.sound only. 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And if the sets A_x are closed of measure < 1, must there be an independent set of size 3? This repository formalizes both answers. Second question: yes — closed sets of measure < 1 admit an infinite independent set (Newelski–Pawlikowski–Seredyński 1987; no boundedness is needed), hence one of size 3. First question: independent of ZFC. Under CH (as ℵ₁ = 𝔠) Hechler's 1972 construction gives a family of bounded sets of outer measure < 1 (in fact countable and null) with no infinite independent set — a Mathlib-level theorem — and the negation of the first question holds in the Boolean-valued model of the collapse algebra Col(ω₁, 𝒫(ω)); after adding 𝔠⁺ random reals (Glazer and Sol 2026, adapting the paper's ω₂ random reals over a CH ground) the first question has a positive answer, so it holds in the Boolean-valued model of the random algebra with 𝔠⁺ coordinates. Both Boolean-valued models are built in a Lean 4 port of Han–van Doorn's Flypitch (the framework of their independence-of-CH proof), and Flypitch's completeness theorem turns them into two-valued models. The comparator challenge states the results in Mathlib alone: the language of set theory, the theory ZFC (Flypitch's axiomatization: extensionality, empty set, ordered pairs, union, power set, infinity, regularity, Zorn's lemma, strong collection) and the sentence Erdos501 (\"every complete ordered field has the Erdős property\") are defined in Mathlib's ModelTheory, and independence is stated semantically as ¬ (ZFC ⊨ᵇ Erdos501) ∧ ¬ (ZFC ⊨ᵇ ∼Erdos501); a faithfulness theorem shows that in Mathlib's ZFSet the sentence is equivalent to the Mathlib statement of the first question, verbatim the proposition of google-deepmind/formal-conjectures. All targets are proved from propext, Classical.choice and Quot.sound only. As far as we know this is the first Erdős problem whose resolution is a formally verified independence result.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000002", "classification": { "arxiv": [ "math.LO", "math.CO" ], "msc2020": [ "03E35", "03E75", "03E40", "03E50", "05D10", "28A05" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000002:erdos501_not_refutable", "name": "erdos501_not_refutable", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000002", "entry_name": "elliotglazer/erdos501", "repository": "https://github.com/elliotglazer/erdos501", "commit": "218d1c1e46f77d4db80e566d1721782e85b94a17", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Erdős problem #501 (Erdős 1961; Erdős–Hajnal 1971, Problem 38): for every real x let A_x be a bounded set of reals of Lebesgue outer measure < 1; must there be an infinite independent set, i.e. an infinite X ⊆ ℝ with x ∉ A_y for all distinct x, y ∈ X? And if the sets A_x are closed of measure < 1, must there be an independent set of size 3? This repository formalizes both answers. Second question: yes — closed sets of measure < 1 admit an infinite independent set (Newelski–Pawlikowski–Seredyński 1987; no boundedness is needed), hence one of size 3. First question: independent of ZFC. Under CH (as ℵ₁ = 𝔠) Hechler's 1972 construction gives a family of bounded sets of outer measure < 1 (in fact countable and null) with no infinite independent set — a Mathlib-level theorem — and the negation of the first question holds in the Boolean-valued model of the collapse algebra Col(ω₁, 𝒫(ω)); after adding 𝔠⁺ random reals (Glazer and Sol 2026, adapting the paper's ω₂ random reals over a CH ground) the first question has a positive answer, so it holds in the Boolean-valued model of the random algebra with 𝔠⁺ coordinates. Both Boolean-valued models are built in a Lean 4 port of Han–van Doorn's Flypitch (the framework of their independence-of-CH proof), and Flypitch's completeness theorem turns them into two-valued models. The comparator challenge states the results in Mathlib alone: the language of set theory, the theory ZFC (Flypitch's axiomatization: extensionality, empty set, ordered pairs, union, power set, infinity, regularity, Zorn's lemma, strong collection) and the sentence Erdos501 (\"every complete ordered field has the Erdős property\") are defined in Mathlib's ModelTheory, and independence is stated semantically as ¬ (ZFC ⊨ᵇ Erdos501) ∧ ¬ (ZFC ⊨ᵇ ∼Erdos501); a faithfulness theorem shows that in Mathlib's ZFSet the sentence is equivalent to the Mathlib statement of the first question, verbatim the proposition of google-deepmind/formal-conjectures. All targets are proved from propext, Classical.choice and Quot.sound only. 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And if the sets A_x are closed of measure < 1, must there be an independent set of size 3? This repository formalizes both answers. Second question: yes — closed sets of measure < 1 admit an infinite independent set (Newelski–Pawlikowski–Seredyński 1987; no boundedness is needed), hence one of size 3. First question: independent of ZFC. Under CH (as ℵ₁ = 𝔠) Hechler's 1972 construction gives a family of bounded sets of outer measure < 1 (in fact countable and null) with no infinite independent set — a Mathlib-level theorem — and the negation of the first question holds in the Boolean-valued model of the collapse algebra Col(ω₁, 𝒫(ω)); after adding 𝔠⁺ random reals (Glazer and Sol 2026, adapting the paper's ω₂ random reals over a CH ground) the first question has a positive answer, so it holds in the Boolean-valued model of the random algebra with 𝔠⁺ coordinates. Both Boolean-valued models are built in a Lean 4 port of Han–van Doorn's Flypitch (the framework of their independence-of-CH proof), and Flypitch's completeness theorem turns them into two-valued models. The comparator challenge states the results in Mathlib alone: the language of set theory, the theory ZFC (Flypitch's axiomatization: extensionality, empty set, ordered pairs, union, power set, infinity, regularity, Zorn's lemma, strong collection) and the sentence Erdos501 (\"every complete ordered field has the Erdős property\") are defined in Mathlib's ModelTheory, and independence is stated semantically as ¬ (ZFC ⊨ᵇ Erdos501) ∧ ¬ (ZFC ⊨ᵇ ∼Erdos501); a faithfulness theorem shows that in Mathlib's ZFSet the sentence is equivalent to the Mathlib statement of the first question, verbatim the proposition of google-deepmind/formal-conjectures. All targets are proved from propext, Classical.choice and Quot.sound only. 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And if the sets A_x are closed of measure < 1, must there be an independent set of size 3? This repository formalizes both answers. Second question: yes — closed sets of measure < 1 admit an infinite independent set (Newelski–Pawlikowski–Seredyński 1987; no boundedness is needed), hence one of size 3. First question: independent of ZFC. Under CH (as ℵ₁ = 𝔠) Hechler's 1972 construction gives a family of bounded sets of outer measure < 1 (in fact countable and null) with no infinite independent set — a Mathlib-level theorem — and the negation of the first question holds in the Boolean-valued model of the collapse algebra Col(ω₁, 𝒫(ω)); after adding 𝔠⁺ random reals (Glazer and Sol 2026, adapting the paper's ω₂ random reals over a CH ground) the first question has a positive answer, so it holds in the Boolean-valued model of the random algebra with 𝔠⁺ coordinates. Both Boolean-valued models are built in a Lean 4 port of Han–van Doorn's Flypitch (the framework of their independence-of-CH proof), and Flypitch's completeness theorem turns them into two-valued models. The comparator challenge states the results in Mathlib alone: the language of set theory, the theory ZFC (Flypitch's axiomatization: extensionality, empty set, ordered pairs, union, power set, infinity, regularity, Zorn's lemma, strong collection) and the sentence Erdos501 (\"every complete ordered field has the Erdős property\") are defined in Mathlib's ModelTheory, and independence is stated semantically as ¬ (ZFC ⊨ᵇ Erdos501) ∧ ¬ (ZFC ⊨ᵇ ∼Erdos501); a faithfulness theorem shows that in Mathlib's ZFSet the sentence is equivalent to the Mathlib statement of the first question, verbatim the proposition of google-deepmind/formal-conjectures. All targets are proved from propext, Classical.choice and Quot.sound only. 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And if the sets A_x are closed of measure < 1, must there be an independent set of size 3? This repository formalizes both answers. Second question: yes — closed sets of measure < 1 admit an infinite independent set (Newelski–Pawlikowski–Seredyński 1987; no boundedness is needed), hence one of size 3. First question: independent of ZFC. Under CH (as ℵ₁ = 𝔠) Hechler's 1972 construction gives a family of bounded sets of outer measure < 1 (in fact countable and null) with no infinite independent set — a Mathlib-level theorem — and the negation of the first question holds in the Boolean-valued model of the collapse algebra Col(ω₁, 𝒫(ω)); after adding 𝔠⁺ random reals (Glazer and Sol 2026, adapting the paper's ω₂ random reals over a CH ground) the first question has a positive answer, so it holds in the Boolean-valued model of the random algebra with 𝔠⁺ coordinates. Both Boolean-valued models are built in a Lean 4 port of Han–van Doorn's Flypitch (the framework of their independence-of-CH proof), and Flypitch's completeness theorem turns them into two-valued models. The comparator challenge states the results in Mathlib alone: the language of set theory, the theory ZFC (Flypitch's axiomatization: extensionality, empty set, ordered pairs, union, power set, infinity, regularity, Zorn's lemma, strong collection) and the sentence Erdos501 (\"every complete ordered field has the Erdős property\") are defined in Mathlib's ModelTheory, and independence is stated semantically as ¬ (ZFC ⊨ᵇ Erdos501) ∧ ¬ (ZFC ⊨ᵇ ∼Erdos501); a faithfulness theorem shows that in Mathlib's ZFSet the sentence is equivalent to the Mathlib statement of the first question, verbatim the proposition of google-deepmind/formal-conjectures. All targets are proved from propext, Classical.choice and Quot.sound only. 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A complex-valued graph parameter that is normalized at the empty graph and invariant under isomorphism has exponentially bounded edge-connection rank if and only if it is the partition function of a mixed edge-colouring model; and when the rank base is R the model may be taken with k even and ℓ odd colours satisfying k, 2ℓ ≤ ⌊2eR⌋. Nothing is assumed. Deligne's theorem on tensor categories, on which the forward direction rests, is itself proved in the development — in the fibre-functor form that is the substance of Deligne's own proof, and by a route through simple algebras in the ind-completion rather than his descent argument — and each of its hypotheses is discharged for a concretely constructed rigid symmetric monoidal envelope of the parameter's skein category. The symmetric-group character theory the argument needs is proved from Mathlib's linear algebra rather than cited, and the converse direction reproves Regts and Sevenster's own theorem. Every theorem of record depends only on propext, Classical.choice and Quot.sound, pinned in-tree so that a change to the axiom set is a build failure rather than a review task.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000003", "classification": { "arxiv": [ "math.CO", "math.CT" ], "msc2020": [ "05C31", "05C50", "18M05", "18M25", "20C30", "68V20" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000003:Certified.regts_sevenster_quant", "name": "Certified.regts_sevenster_quant", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000003", "entry_name": "WillWhistler/Regts-Sevenster", "repository": "https://github.com/WillWhistler/Regts-Sevenster", "commit": "504ec6aada68b4472f32c4ad46452ff86614f732", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalization of the Regts–Sevenster conjecture, proved in both directions and resting on no assumption. A complex-valued graph parameter that is normalized at the empty graph and invariant under isomorphism has exponentially bounded edge-connection rank if and only if it is the partition function of a mixed edge-colouring model; and when the rank base is R the model may be taken with k even and ℓ odd colours satisfying k, 2ℓ ≤ ⌊2eR⌋. Nothing is assumed. Deligne's theorem on tensor categories, on which the forward direction rests, is itself proved in the development — in the fibre-functor form that is the substance of Deligne's own proof, and by a route through simple algebras in the ind-completion rather than his descent argument — and each of its hypotheses is discharged for a concretely constructed rigid symmetric monoidal envelope of the parameter's skein category. The symmetric-group character theory the argument needs is proved from Mathlib's linear algebra rather than cited, and the converse direction reproves Regts and Sevenster's own theorem. Every theorem of record depends only on propext, Classical.choice and Quot.sound, pinned in-tree so that a change to the axiom set is a build failure rather than a review task.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000003", "classification": { "arxiv": [ "math.CO", "math.CT" ], "msc2020": [ "05C31", "05C50", "18M05", "18M25", "20C30", "68V20" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000003:Certified.regts_sevenster_converse", "name": "Certified.regts_sevenster_converse", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000003", "entry_name": "WillWhistler/Regts-Sevenster", "repository": "https://github.com/WillWhistler/Regts-Sevenster", "commit": "504ec6aada68b4472f32c4ad46452ff86614f732", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalization of the Regts–Sevenster conjecture, proved in both directions and resting on no assumption. A complex-valued graph parameter that is normalized at the empty graph and invariant under isomorphism has exponentially bounded edge-connection rank if and only if it is the partition function of a mixed edge-colouring model; and when the rank base is R the model may be taken with k even and ℓ odd colours satisfying k, 2ℓ ≤ ⌊2eR⌋. Nothing is assumed. Deligne's theorem on tensor categories, on which the forward direction rests, is itself proved in the development — in the fibre-functor form that is the substance of Deligne's own proof, and by a route through simple algebras in the ind-completion rather than his descent argument — and each of its hypotheses is discharged for a concretely constructed rigid symmetric monoidal envelope of the parameter's skein category. The symmetric-group character theory the argument needs is proved from Mathlib's linear algebra rather than cited, and the converse direction reproves Regts and Sevenster's own theorem. Every theorem of record depends only on propext, Classical.choice and Quot.sound, pinned in-tree so that a change to the axiom set is a build failure rather than a review task.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000003", "classification": { "arxiv": [ "math.CO", "math.CT" ], "msc2020": [ "05C31", "05C50", "18M05", "18M25", "20C30", "68V20" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000003:Certified.edgeRankBounded_of_mixedBounded", "name": "Certified.edgeRankBounded_of_mixedBounded", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000003", "entry_name": "WillWhistler/Regts-Sevenster", "repository": "https://github.com/WillWhistler/Regts-Sevenster", "commit": "504ec6aada68b4472f32c4ad46452ff86614f732", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalization of the Regts–Sevenster conjecture, proved in both directions and resting on no assumption. A complex-valued graph parameter that is normalized at the empty graph and invariant under isomorphism has exponentially bounded edge-connection rank if and only if it is the partition function of a mixed edge-colouring model; and when the rank base is R the model may be taken with k even and ℓ odd colours satisfying k, 2ℓ ≤ ⌊2eR⌋. Nothing is assumed. Deligne's theorem on tensor categories, on which the forward direction rests, is itself proved in the development — in the fibre-functor form that is the substance of Deligne's own proof, and by a route through simple algebras in the ind-completion rather than his descent argument — and each of its hypotheses is discharged for a concretely constructed rigid symmetric monoidal envelope of the parameter's skein category. The symmetric-group character theory the argument needs is proved from Mathlib's linear algebra rather than cited, and the converse direction reproves Regts and Sevenster's own theorem. Every theorem of record depends only on propext, Classical.choice and Quot.sound, pinned in-tree so that a change to the axiom set is a build failure rather than a review task.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000003", "classification": { "arxiv": [ "math.CO", "math.CT" ], "msc2020": [ "05C31", "05C50", "18M05", "18M25", "20C30", "68V20" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000003:Certified.regts_sevenster_characterisation", "name": "Certified.regts_sevenster_characterisation", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000003", "entry_name": "WillWhistler/Regts-Sevenster", "repository": "https://github.com/WillWhistler/Regts-Sevenster", "commit": "504ec6aada68b4472f32c4ad46452ff86614f732", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalization of the Regts–Sevenster conjecture, proved in both directions and resting on no assumption. A complex-valued graph parameter that is normalized at the empty graph and invariant under isomorphism has exponentially bounded edge-connection rank if and only if it is the partition function of a mixed edge-colouring model; and when the rank base is R the model may be taken with k even and ℓ odd colours satisfying k, 2ℓ ≤ ⌊2eR⌋. Nothing is assumed. Deligne's theorem on tensor categories, on which the forward direction rests, is itself proved in the development — in the fibre-functor form that is the substance of Deligne's own proof, and by a route through simple algebras in the ind-completion rather than his descent argument — and each of its hypotheses is discharged for a concretely constructed rigid symmetric monoidal envelope of the parameter's skein category. The symmetric-group character theory the argument needs is proved from Mathlib's linear algebra rather than cited, and the converse direction reproves Regts and Sevenster's own theorem. Every theorem of record depends only on propext, Classical.choice and Quot.sound, pinned in-tree so that a change to the axiom set is a build failure rather than a review task.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000003", "classification": { "arxiv": [ "math.CO", "math.CT" ], "msc2020": [ "05C31", "05C50", "18M05", "18M25", "20C30", "68V20" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000003:Certified.regts_sevenster_quant_characterisation", "name": "Certified.regts_sevenster_quant_characterisation", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000003", "entry_name": "WillWhistler/Regts-Sevenster", "repository": "https://github.com/WillWhistler/Regts-Sevenster", "commit": "504ec6aada68b4472f32c4ad46452ff86614f732", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalization of the Regts–Sevenster conjecture, proved in both directions and resting on no assumption. A complex-valued graph parameter that is normalized at the empty graph and invariant under isomorphism has exponentially bounded edge-connection rank if and only if it is the partition function of a mixed edge-colouring model; and when the rank base is R the model may be taken with k even and ℓ odd colours satisfying k, 2ℓ ≤ ⌊2eR⌋. Nothing is assumed. Deligne's theorem on tensor categories, on which the forward direction rests, is itself proved in the development — in the fibre-functor form that is the substance of Deligne's own proof, and by a route through simple algebras in the ind-completion rather than his descent argument — and each of its hypotheses is discharged for a concretely constructed rigid symmetric monoidal envelope of the parameter's skein category. The symmetric-group character theory the argument needs is proved from Mathlib's linear algebra rather than cited, and the converse direction reproves Regts and Sevenster's own theorem. Every theorem of record depends only on propext, Classical.choice and Quot.sound, pinned in-tree so that a change to the axiom set is a build failure rather than a review task.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000003", "classification": { "arxiv": [ "math.CO", "math.CT" ], "msc2020": [ "05C31", "05C50", "18M05", "18M25", "20C30", "68V20" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000004:Wallace.Palomar.torsionFreeAbelianGroup_mainTheorem", "name": "Wallace.Palomar.torsionFreeAbelianGroup_mainTheorem", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000004", "entry_name": "vo-rodrigues/wallace-problem-palomar", "repository": "https://github.com/vo-rodrigues/wallace-problem-palomar", "commit": "1ab2fb7ed1fb106e14af45312c2a7ab5d568048b", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Lean 4 formalization of the theorem that every torsion-free Abelian group of cardinality continuum admits a Hausdorff countably compact group topology in which every convergent sequence is eventually constant, and constructs a commutative Tychonoff countably compact cancellative topological monoid that is not a group, giving a negative answer to Wallace's question.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000004", "classification": { "arxiv": [ "math.GR", "math.GN" ], "msc2020": [ "22A05", "54H11", "54D20", "22A15", "20K20", "54A20" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000004:Wallace.Palomar.commutativeTychonoffWallaceCounterexampleExists", "name": "Wallace.Palomar.commutativeTychonoffWallaceCounterexampleExists", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000004", "entry_name": "vo-rodrigues/wallace-problem-palomar", "repository": "https://github.com/vo-rodrigues/wallace-problem-palomar", "commit": "1ab2fb7ed1fb106e14af45312c2a7ab5d568048b", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Lean 4 formalization of the theorem that every torsion-free Abelian group of cardinality continuum admits a Hausdorff countably compact group topology in which every convergent sequence is eventually constant, and constructs a commutative Tychonoff countably compact cancellative topological monoid that is not a group, giving a negative answer to Wallace's question.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000004", "classification": { "arxiv": [ "math.GR", "math.GN" ], "msc2020": [ "22A05", "54H11", "54D20", "22A15", "20K20", "54A20" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000005:TheoremA", "name": "TheoremA", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000005", "entry_name": "dcposch/jc2-lean", "repository": "https://github.com/dcposch/jc2-lean", "commit": "6c0f56309226432afb90b6213638ec987d46f4a3", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Lean 4 / Mathlib formalization of Theorem A (ODE rigidity, all weights; label thm:ode, Theorem 6.1 in the current compilation of the campaign paper): if polynomials A, C over an integral domain of characteristic zero and an integer weight nu >= 1 satisfy A*C' - nu*A'*C = c for a nonzero constant c, then deg A <= 1 (TheoremA). Also a standalone positive-characteristic strengthening extracted from the same proof (TheoremA_charP): the same conclusion over a domain of characteristic p under the degree bounds nu*deg A < p and deg C < p. The theorem is the rigidity engine of the depth-two block analysis in the campaign on the plane Jacobian problem at Newton-degree pair (72,108).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000005", "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000005:TheoremA_charP", "name": "TheoremA_charP", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000005", "entry_name": "dcposch/jc2-lean", "repository": "https://github.com/dcposch/jc2-lean", "commit": "6c0f56309226432afb90b6213638ec987d46f4a3", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Lean 4 / Mathlib formalization of Theorem A (ODE rigidity, all weights; label thm:ode, Theorem 6.1 in the current compilation of the campaign paper): if polynomials A, C over an integral domain of characteristic zero and an integer weight nu >= 1 satisfy A*C' - nu*A'*C = c for a nonzero constant c, then deg A <= 1 (TheoremA). Also a standalone positive-characteristic strengthening extracted from the same proof (TheoremA_charP): the same conclusion over a domain of characteristic p under the degree bounds nu*deg A < p and deg C < p. The theorem is the rigidity engine of the depth-two block analysis in the campaign on the plane Jacobian problem at Newton-degree pair (72,108).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000005", "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000006:ArithmeticOfTime.padovan_a_forces_b", "name": "ArithmeticOfTime.padovan_a_forces_b", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000006", "entry_name": "stalex444/arithmetic-of-time", "repository": "https://github.com/stalex444/arithmetic-of-time", "commit": "c48eb12817dd1d7ff3dc4d92306c7a4f249e6d7d", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A kernel-checked chirality package for the Padovan substitution (0 ↦ 1, 1 ↦ 2, 2 ↦ 01 on Fin 3; its incidence matrix has characteristic polynomial x³ − x − 1, an orientation fact that is not used or compared). Three compared theorems, all at the iterate-factor level — they quantify over factors of the concrete words S^[n] [x], never over an abstract subshift. (1) The successor invariant: in every positive iterate from any seed, the letter 0 (\"a\") is immediately followed by 1 (\"b\"). (2) Chirality: the word 20 (\"ca\") is a factor of an iterate while its reversal 02 (\"ac\") is a factor of no iterate from any seed — the factor language reads differently backwards — proved from the invariant, with no enumeration on the ac side. (3) No palindromic factor: no factor of length ≥ 4 of any iterate from any seed (including the zeroth iterate) is a palindrome, via a complete length-≤5 window certificate for the seed a together with the index shift S [0] = [1], S [1] = [2], which exhibits every seed's iterate as a seed-a iterate.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000006", "classification": { "arxiv": [ "math.CO", "math.DS" ], "msc2020": [ "68R15", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000006:ArithmeticOfTime.padovan_chirality", "name": "ArithmeticOfTime.padovan_chirality", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000006", "entry_name": "stalex444/arithmetic-of-time", "repository": "https://github.com/stalex444/arithmetic-of-time", "commit": "c48eb12817dd1d7ff3dc4d92306c7a4f249e6d7d", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A kernel-checked chirality package for the Padovan substitution (0 ↦ 1, 1 ↦ 2, 2 ↦ 01 on Fin 3; its incidence matrix has characteristic polynomial x³ − x − 1, an orientation fact that is not used or compared). Three compared theorems, all at the iterate-factor level — they quantify over factors of the concrete words S^[n] [x], never over an abstract subshift. (1) The successor invariant: in every positive iterate from any seed, the letter 0 (\"a\") is immediately followed by 1 (\"b\"). (2) Chirality: the word 20 (\"ca\") is a factor of an iterate while its reversal 02 (\"ac\") is a factor of no iterate from any seed — the factor language reads differently backwards — proved from the invariant, with no enumeration on the ac side. (3) No palindromic factor: no factor of length ≥ 4 of any iterate from any seed (including the zeroth iterate) is a palindrome, via a complete length-≤5 window certificate for the seed a together with the index shift S [0] = [1], S [1] = [2], which exhibits every seed's iterate as a seed-a iterate.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000006", "classification": { "arxiv": [ "math.CO", "math.DS" ], "msc2020": [ "68R15", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000006:ArithmeticOfTime.padovan_no_long_palindrome", "name": "ArithmeticOfTime.padovan_no_long_palindrome", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000006", "entry_name": "stalex444/arithmetic-of-time", "repository": "https://github.com/stalex444/arithmetic-of-time", "commit": "c48eb12817dd1d7ff3dc4d92306c7a4f249e6d7d", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A kernel-checked chirality package for the Padovan substitution (0 ↦ 1, 1 ↦ 2, 2 ↦ 01 on Fin 3; its incidence matrix has characteristic polynomial x³ − x − 1, an orientation fact that is not used or compared). Three compared theorems, all at the iterate-factor level — they quantify over factors of the concrete words S^[n] [x], never over an abstract subshift. (1) The successor invariant: in every positive iterate from any seed, the letter 0 (\"a\") is immediately followed by 1 (\"b\"). (2) Chirality: the word 20 (\"ca\") is a factor of an iterate while its reversal 02 (\"ac\") is a factor of no iterate from any seed — the factor language reads differently backwards — proved from the invariant, with no enumeration on the ac side. (3) No palindromic factor: no factor of length ≥ 4 of any iterate from any seed (including the zeroth iterate) is a palindrome, via a complete length-≤5 window certificate for the seed a together with the index shift S [0] = [1], S [1] = [2], which exhibits every seed's iterate as a seed-a iterate.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000006", "classification": { "arxiv": [ "math.CO", "math.DS" ], "msc2020": [ "68R15", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.chsh_upper", "name": "TsirelsonTightness.chsh_upper", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.chsh_mul_self", "name": "TsirelsonTightness.chsh_mul_self", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.chsh_norm_le", "name": "TsirelsonTightness.chsh_norm_le", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.chsh_norm_of_comm", "name": "TsirelsonTightness.chsh_norm_of_comm", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.noncomm_of_chsh_norm_gt_two", "name": "TsirelsonTightness.noncomm_of_chsh_norm_gt_two", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.tuple_isCHSH", "name": "TsirelsonTightness.tuple_isCHSH", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.vsat_ne_zero", "name": "TsirelsonTightness.vsat_ne_zero", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.chsh_saturates", "name": "TsirelsonTightness.chsh_saturates", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.classical_le_two", "name": "TsirelsonTightness.classical_le_two", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.bell_gap", "name": "TsirelsonTightness.bell_gap", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.chsh_opNorm", "name": "TsirelsonTightness.chsh_opNorm", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.chsh_sq_ne", "name": "TsirelsonTightness.chsh_sq_ne", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-19-000007:TsirelsonTightness.chsh_opNorm_isGreatest", "name": "TsirelsonTightness.chsh_opNorm_isGreatest", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-19-000007", "entry_name": "stalex444/pdt-lean", "repository": "https://github.com/stalex444/pdt-lean", "commit": "0dae307fb786a3d180bba137a44555a1bf645637", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Kernel-checked exact-constant results for the CHSH operator T = A₀B₀ + A₀B₁ + A₁B₀ − A₁B₁ of an arbitrary CHSH tuple (Mathlib's IsCHSHTuple: self-adjoint involutions with the A's commuting with the B's). (1) The Landau identity: in any *-ring, T² = 4 + [A₀,A₁]·[B₁,B₀] — the exact, representation-independent correction to the naive T² = 8·1 (chsh_mul_self). (2) Tsirelson's bound in norm form: ‖T‖ ≤ 2√2 for every CHSH tuple in every nontrivial unital real C*-normed algebra (chsh_norm_le). (3) The classical constant, exactly: if either party's pair commutes, ‖T‖ = 2 — not merely ≤ 2 (chsh_norm_of_comm). (4) Necessity of noncommutativity: any violation ‖T‖ > 2 forces A₀A₁ ≠ A₁A₀ and B₀B₁ ≠ B₁B₀ — noncommutativity within both parties is necessary for any violation of the classical bound (noncomm_of_chsh_norm_gt_two). (5) The exact constant of M₄(ℝ): in the l2 operator norm, 2√2 is the greatest element of the set of CHSH operator norms over 4×4 real matrices — an upper bound for every tuple, attained by an explicit real-Pauli tuple (chsh_opNorm_isGreatest), so the maximal violation 2√2 is possible, not just bounded. The witness side stays compared: the explicit quadruple is verified to be a genuine CHSH tuple, its CHSH operator has eigenvalue exactly 2√2 on the explicit nonzero vector (1,0,0,1) and l2 operator norm exactly 2√2; the ordered star-ring bounds T ≤ (2√2)·1 and (commutative case) T ≤ 2 and the strict gap 2 < 2√2 complete the picture. A falsification is compared alongside: the saturating tuple's T² is not c·1 for ANY real c (chsh_sq_ne quantifies over every scalar), so the scalar-square route to the norm is closed for every scalar; the norm results here go through the C*-identity (and, for the generic bound, the Landau identity) instead.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-19-000007", "classification": { "arxiv": [ "quant-ph", "math.OA" ], "msc2020": [ "81P40", "81P45", "46L60", "15A18" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000001:CornerEnumeration", "name": "CornerEnumeration", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000001", "entry_name": "dcposch/jc2-lean", "repository": "https://github.com/dcposch/jc2-lean", "commit": "3cee8100665bc59734511500b98728628ea190db", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Lean 4 / Mathlib formalization of the vertex-gap obstruction at (k,d2) = (2,2) (label thm:22, Theorem 3.4 in the current compilation) from the campaign paper \"A Vertex-Gap Obstruction for Low-Degree Strip Pairs in the Plane Jacobian Conjecture\": for polynomials P, Q in K[x,y] over a field K of characteristic not in {2,3,5}, supported in the normalized type-(2,2) Newton strips along (1,2) with [P,Q] = x^2, the block coefficients satisfy a3 = 0 and a2^2*a6 = 0 - so the generic chart a2*a6 != 0 of the strip family is empty (VertexGap22, plus the side-symmetric companion VertexGap22_swapped in the mirrored labels). Also formalized: the arithmetic corner-enumeration lemma behind the vertex normalization (lem:enum, Lemma 2.2; CornerEnumeration) and the side-symmetric form of the gap condition, hypothesis (ii) as corrected by the erratum recorded in Section 8 (GapConditionSideSymm). The theorem is the mechanism that empties the generic chart of the (8,28) strip subcase in the campaign on the plane Jacobian problem at Newton-degree pair (72,108).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000001", "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000001:GapConditionSideSymm", "name": "GapConditionSideSymm", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000001", "entry_name": "dcposch/jc2-lean", "repository": "https://github.com/dcposch/jc2-lean", "commit": "3cee8100665bc59734511500b98728628ea190db", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Lean 4 / Mathlib formalization of the vertex-gap obstruction at (k,d2) = (2,2) (label thm:22, Theorem 3.4 in the current compilation) from the campaign paper \"A Vertex-Gap Obstruction for Low-Degree Strip Pairs in the Plane Jacobian Conjecture\": for polynomials P, Q in K[x,y] over a field K of characteristic not in {2,3,5}, supported in the normalized type-(2,2) Newton strips along (1,2) with [P,Q] = x^2, the block coefficients satisfy a3 = 0 and a2^2*a6 = 0 - so the generic chart a2*a6 != 0 of the strip family is empty (VertexGap22, plus the side-symmetric companion VertexGap22_swapped in the mirrored labels). Also formalized: the arithmetic corner-enumeration lemma behind the vertex normalization (lem:enum, Lemma 2.2; CornerEnumeration) and the side-symmetric form of the gap condition, hypothesis (ii) as corrected by the erratum recorded in Section 8 (GapConditionSideSymm). The theorem is the mechanism that empties the generic chart of the (8,28) strip subcase in the campaign on the plane Jacobian problem at Newton-degree pair (72,108).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000001", "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000001:VertexGap22", "name": "VertexGap22", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000001", "entry_name": "dcposch/jc2-lean", "repository": "https://github.com/dcposch/jc2-lean", "commit": "3cee8100665bc59734511500b98728628ea190db", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Lean 4 / Mathlib formalization of the vertex-gap obstruction at (k,d2) = (2,2) (label thm:22, Theorem 3.4 in the current compilation) from the campaign paper \"A Vertex-Gap Obstruction for Low-Degree Strip Pairs in the Plane Jacobian Conjecture\": for polynomials P, Q in K[x,y] over a field K of characteristic not in {2,3,5}, supported in the normalized type-(2,2) Newton strips along (1,2) with [P,Q] = x^2, the block coefficients satisfy a3 = 0 and a2^2*a6 = 0 - so the generic chart a2*a6 != 0 of the strip family is empty (VertexGap22, plus the side-symmetric companion VertexGap22_swapped in the mirrored labels). Also formalized: the arithmetic corner-enumeration lemma behind the vertex normalization (lem:enum, Lemma 2.2; CornerEnumeration) and the side-symmetric form of the gap condition, hypothesis (ii) as corrected by the erratum recorded in Section 8 (GapConditionSideSymm). The theorem is the mechanism that empties the generic chart of the (8,28) strip subcase in the campaign on the plane Jacobian problem at Newton-degree pair (72,108).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000001", "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000001:VertexGap22_swapped", "name": "VertexGap22_swapped", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000001", "entry_name": "dcposch/jc2-lean", "repository": "https://github.com/dcposch/jc2-lean", "commit": "3cee8100665bc59734511500b98728628ea190db", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Lean 4 / Mathlib formalization of the vertex-gap obstruction at (k,d2) = (2,2) (label thm:22, Theorem 3.4 in the current compilation) from the campaign paper \"A Vertex-Gap Obstruction for Low-Degree Strip Pairs in the Plane Jacobian Conjecture\": for polynomials P, Q in K[x,y] over a field K of characteristic not in {2,3,5}, supported in the normalized type-(2,2) Newton strips along (1,2) with [P,Q] = x^2, the block coefficients satisfy a3 = 0 and a2^2*a6 = 0 - so the generic chart a2*a6 != 0 of the strip family is empty (VertexGap22, plus the side-symmetric companion VertexGap22_swapped in the mirrored labels). Also formalized: the arithmetic corner-enumeration lemma behind the vertex normalization (lem:enum, Lemma 2.2; CornerEnumeration) and the side-symmetric form of the gap condition, hypothesis (ii) as corrected by the erratum recorded in Section 8 (GapConditionSideSymm). The theorem is the mechanism that empties the generic chart of the (8,28) strip subcase in the campaign on the plane Jacobian problem at Newton-degree pair (72,108).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000001", "classification": { "arxiv": [ "math.AG", "math.AC" ], "msc2020": [ "14R15" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000002:LowRankUnivariateSOS.rankTwo_no_spurious_socp", "name": "LowRankUnivariateSOS.rankTwo_no_spurious_socp", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000002", "entry_name": "yuanchenyang/lean_low_rank_univariate_sos", "repository": "https://github.com/yuanchenyang/lean_low_rank_univariate_sos", "commit": "1fccd921a4530a088ab6e230f832041ab3c5c7f3", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalization of the main theorem of Legat, Yuan and Parrilo, \"Low-Rank Univariate Sum of Squares Has No Spurious Local Minima\" (SIAM J. Optim. 33(3), 2023). For a real univariate polynomial p and a pair of polynomials u = (u₁, u₂), consider the quadratically penalized Burer–Monteiro objective f_p(u) = ‖σ(u) - p‖² with σ(u) = u₁² + u₂², the norm coming from any inner product on ℝ[x]. The formalized theorem states that if p is a sum of two squares (which is the case for any nonnegative univariate polynomial) and u is a second-order critical point of f_p (vanishing gradient and positive semidefinite Hessian, Definition 3.1 of the paper), then σ(u) = p, equivalently f_p(u) = 0; hence every SOCP, and in particular every local minimum, is a global minimum. The development follows the paper's proof: the gcd decomposition u = a·u₀ with u₀ coprime, the observation that σ(u₀) has no real roots, a Bézout-based \"hgroup\" lemma writing a sum of squares p as g·q + s·σ(u₀) with s a sum of squares when g and σ(u₀) are coprime, the factor-peeling argument that replaces the quadratic factors of a shared with σ(u₀) by X², and the final certificate argument combining the first- and second-order conditions.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000002", "classification": { "arxiv": [ "math.OC", "math.AG" ], "msc2020": [ "90C23", "90C26", "90C22", "12D15" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000003:card_valid_bases", "name": "card_valid_bases", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000003", "entry_name": "catskillsresearch/cardb", "repository": "https://github.com/catskillsresearch/cardb", "commit": "d8c1d63052a18db0c43d44c1c77fb10ca8902ed0", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean formalization of an exact enumeration identity and two consequences: for a finite type α, the number of families B ⊆ Set α that are topological bases equals the sum, over all topologies τ on α, of 2^(|τ| − |M_τ|), where M_τ is the canonical minimal basis of minimal open neighbourhoods (Mathlib's nhdsKer {x} at each point). Kernel-reducible enumeration gives the exact table #(0)=2, #(1)=2, #(2)=10, #(3)=142. A quantitative sandwich then shows that the discrete fiber 2^(2^N − N) dominates: for N ≥ 2 the remaining fibers contribute at most 2^(N(N−1) + 3·2^(N−2)), the ratio to the discrete term tends to 1, and for N ≥ 10 the total is at most twice the discrete term. The argument is a fiber decomposition: every finite topology is Alexandrov-discrete, the minimal neighbourhoods generate it, and the bases generating τ are exactly the families B with M_τ ⊆ B ⊆ {U | τ.IsOpen U}. The informal account is the companion note CARDB.md in this repository. The Lean source was written by AI agents under the direction and review of the author. 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Kernel-reducible enumeration gives the exact table #(0)=2, #(1)=2, #(2)=10, #(3)=142. A quantitative sandwich then shows that the discrete fiber 2^(2^N − N) dominates: for N ≥ 2 the remaining fibers contribute at most 2^(N(N−1) + 3·2^(N−2)), the ratio to the discrete term tends to 1, and for N ≥ 10 the total is at most twice the discrete term. The argument is a fiber decomposition: every finite topology is Alexandrov-discrete, the minimal neighbourhoods generate it, and the bases generating τ are exactly the families B with M_τ ⊆ B ⊆ {U | τ.IsOpen U}. The informal account is the companion note CARDB.md in this repository. The Lean source was written by AI agents under the direction and review of the author. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000003", "classification": { "arxiv": [ "math.GN", "math.CO" ], "msc2020": [ "54A05", "05A15", "54A10" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000003:card_valid_bases_bounds", "name": "card_valid_bases_bounds", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000003", "entry_name": "catskillsresearch/cardb", "repository": "https://github.com/catskillsresearch/cardb", "commit": "d8c1d63052a18db0c43d44c1c77fb10ca8902ed0", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean formalization of an exact enumeration identity and two consequences: for a finite type α, the number of families B ⊆ Set α that are topological bases equals the sum, over all topologies τ on α, of 2^(|τ| − |M_τ|), where M_τ is the canonical minimal basis of minimal open neighbourhoods (Mathlib's nhdsKer {x} at each point). Kernel-reducible enumeration gives the exact table #(0)=2, #(1)=2, #(2)=10, #(3)=142. A quantitative sandwich then shows that the discrete fiber 2^(2^N − N) dominates: for N ≥ 2 the remaining fibers contribute at most 2^(N(N−1) + 3·2^(N−2)), the ratio to the discrete term tends to 1, and for N ≥ 10 the total is at most twice the discrete term. The argument is a fiber decomposition: every finite topology is Alexandrov-discrete, the minimal neighbourhoods generate it, and the bases generating τ are exactly the families B with M_τ ⊆ B ⊆ {U | τ.IsOpen U}. The informal account is the companion note CARDB.md in this repository. The Lean source was written by AI agents under the direction and review of the author. 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Kernel-reducible enumeration gives the exact table #(0)=2, #(1)=2, #(2)=10, #(3)=142. A quantitative sandwich then shows that the discrete fiber 2^(2^N − N) dominates: for N ≥ 2 the remaining fibers contribute at most 2^(N(N−1) + 3·2^(N−2)), the ratio to the discrete term tends to 1, and for N ≥ 10 the total is at most twice the discrete term. The argument is a fiber decomposition: every finite topology is Alexandrov-discrete, the minimal neighbourhoods generate it, and the bases generating τ are exactly the families B with M_τ ⊆ B ⊆ {U | τ.IsOpen U}. The informal account is the companion note CARDB.md in this repository. The Lean source was written by AI agents under the direction and review of the author. 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Kernel-reducible enumeration gives the exact table #(0)=2, #(1)=2, #(2)=10, #(3)=142. A quantitative sandwich then shows that the discrete fiber 2^(2^N − N) dominates: for N ≥ 2 the remaining fibers contribute at most 2^(N(N−1) + 3·2^(N−2)), the ratio to the discrete term tends to 1, and for N ≥ 10 the total is at most twice the discrete term. The argument is a fiber decomposition: every finite topology is Alexandrov-discrete, the minimal neighbourhoods generate it, and the bases generating τ are exactly the families B with M_τ ⊆ B ⊆ {U | τ.IsOpen U}. The informal account is the companion note CARDB.md in this repository. The Lean source was written by AI agents under the direction and review of the author. 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This conjecture was proposed by Shapirov in \"Perfect cuboids and irreducible polynomials\" (arXiv:1108.5348) as part of a family of three conjectures whose combined truth would imply the nonexistence of a perfect Euler brick. The proof first analyses rational factors and the symmetry F(−x) vs F(x) to reduce irreducibility of the degree-8 polynomial to irreducibility of a degree-4 factor Q_{a,b}, and the coprime branch to a Diophantine identity. Irreducibility of Q is in turn reduced to the quartic Diophantine equation x⁴ + 34x²y² + y⁴ = z², whose only trivial solutions are established by Fermat-style infinite descent. The formal proof was autonomously discovered by the AlphaProof Nexus agent; an independent informal proof using rank-zero elliptic curves was given by Asiryan (arXiv:2510.11768).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000005", "classification": { "arxiv": [ "math.NT", "math.AG" ], "msc2020": [ "11D41", "12E05" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000006:SZ.quadPisot_notMem_S_of_two_lt", "name": "SZ.quadPisot_notMem_S_of_two_lt", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000006", "entry_name": "rwst/Even-Integral", "repository": "https://github.com/rwst/Even-Integral", "commit": "db8f6025df35517cb9552df04d18345c68d8ac11", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which improves on a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 5 asks whether 𝒮 has an element greater than 2. This formalization decides 𝒮 versus 𝒵 for every quadratic Pisot number: the golden mean is the only one in 𝒮, so none greater than 2 lies there and Problem 5 loses its whole natural class of candidates, though the problem itself stays open. The case that had resisted the source paper's four constructions is α = 1 + √2, settled by an exact computation rather than an estimate: with ξ = (2−√2)/4 one has ⌊ξ(1+√2)^(n+1)⌋ = Pₙ − [n odd] for the Pell numbers P, so every integral part is even. The mechanism generalizes: for a real quadratic pair with second conjugate in (−1,0), the closed-form multiplier ξ = −β/(α−β) makes m divide ⌊ξαⁿ⌋ for every n ≥ 0 whenever m ∣ a and m ∣ b−1 — an exact congruence, not a confinement of fractional parts — and all but finitely many of those integral parts are then composite. The informal account is the companion paper doi:10.13140/RG.2.2.20582.79689. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery and the formalization are the AI agent's; the agent also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the agent is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000006", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11R06", "11B37", "11B39", "68V20" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000006:SZ.isQuadPisot_silver", "name": "SZ.isQuadPisot_silver", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000006", "entry_name": "rwst/Even-Integral", "repository": "https://github.com/rwst/Even-Integral", "commit": "db8f6025df35517cb9552df04d18345c68d8ac11", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which improves on a problem by Dubickas. 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The mechanism generalizes: for a real quadratic pair with second conjugate in (−1,0), the closed-form multiplier ξ = −β/(α−β) makes m divide ⌊ξαⁿ⌋ for every n ≥ 0 whenever m ∣ a and m ∣ b−1 — an exact congruence, not a confinement of fractional parts — and all but finitely many of those integral parts are then composite. The informal account is the companion paper doi:10.13140/RG.2.2.20582.79689. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery and the formalization are the AI agent's; the agent also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the agent is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000006", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11R06", "11B37", "11B39", "68V20" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000006:SZ.quadPisot_mem_S_iff", "name": "SZ.quadPisot_mem_S_iff", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000006", "entry_name": "rwst/Even-Integral", "repository": "https://github.com/rwst/Even-Integral", "commit": "db8f6025df35517cb9552df04d18345c68d8ac11", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which improves on a problem by Dubickas. 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The mechanism generalizes: for a real quadratic pair with second conjugate in (−1,0), the closed-form multiplier ξ = −β/(α−β) makes m divide ⌊ξαⁿ⌋ for every n ≥ 0 whenever m ∣ a and m ∣ b−1 — an exact congruence, not a confinement of fractional parts — and all but finitely many of those integral parts are then composite. The informal account is the companion paper doi:10.13140/RG.2.2.20582.79689. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery and the formalization are the AI agent's; the agent also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the agent is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000006", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11R06", "11B37", "11B39", "68V20" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000006:SZ.quadPisot_mem_MahlerZ", "name": "SZ.quadPisot_mem_MahlerZ", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000006", "entry_name": "rwst/Even-Integral", "repository": "https://github.com/rwst/Even-Integral", "commit": "db8f6025df35517cb9552df04d18345c68d8ac11", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which improves on a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 5 asks whether 𝒮 has an element greater than 2. This formalization decides 𝒮 versus 𝒵 for every quadratic Pisot number: the golden mean is the only one in 𝒮, so none greater than 2 lies there and Problem 5 loses its whole natural class of candidates, though the problem itself stays open. The case that had resisted the source paper's four constructions is α = 1 + √2, settled by an exact computation rather than an estimate: with ξ = (2−√2)/4 one has ⌊ξ(1+√2)^(n+1)⌋ = Pₙ − [n odd] for the Pell numbers P, so every integral part is even. The mechanism generalizes: for a real quadratic pair with second conjugate in (−1,0), the closed-form multiplier ξ = −β/(α−β) makes m divide ⌊ξαⁿ⌋ for every n ≥ 0 whenever m ∣ a and m ∣ b−1 — an exact congruence, not a confinement of fractional parts — and all but finitely many of those integral parts are then composite. The informal account is the companion paper doi:10.13140/RG.2.2.20582.79689. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery and the formalization are the AI agent's; the agent also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the agent is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. 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Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 5 asks whether 𝒮 has an element greater than 2. This formalization decides 𝒮 versus 𝒵 for every quadratic Pisot number: the golden mean is the only one in 𝒮, so none greater than 2 lies there and Problem 5 loses its whole natural class of candidates, though the problem itself stays open. The case that had resisted the source paper's four constructions is α = 1 + √2, settled by an exact computation rather than an estimate: with ξ = (2−√2)/4 one has ⌊ξ(1+√2)^(n+1)⌋ = Pₙ − [n odd] for the Pell numbers P, so every integral part is even. The mechanism generalizes: for a real quadratic pair with second conjugate in (−1,0), the closed-form multiplier ξ = −β/(α−β) makes m divide ⌊ξαⁿ⌋ for every n ≥ 0 whenever m ∣ a and m ∣ b−1 — an exact congruence, not a confinement of fractional parts — and all but finitely many of those integral parts are then composite. The informal account is the companion paper doi:10.13140/RG.2.2.20582.79689. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery and the formalization are the AI agent's; the agent also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the agent is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. 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Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 5 asks whether 𝒮 has an element greater than 2. This formalization decides 𝒮 versus 𝒵 for every quadratic Pisot number: the golden mean is the only one in 𝒮, so none greater than 2 lies there and Problem 5 loses its whole natural class of candidates, though the problem itself stays open. The case that had resisted the source paper's four constructions is α = 1 + √2, settled by an exact computation rather than an estimate: with ξ = (2−√2)/4 one has ⌊ξ(1+√2)^(n+1)⌋ = Pₙ − [n odd] for the Pell numbers P, so every integral part is even. The mechanism generalizes: for a real quadratic pair with second conjugate in (−1,0), the closed-form multiplier ξ = −β/(α−β) makes m divide ⌊ξαⁿ⌋ for every n ≥ 0 whenever m ∣ a and m ∣ b−1 — an exact congruence, not a confinement of fractional parts — and all but finitely many of those integral parts are then composite. The informal account is the companion paper doi:10.13140/RG.2.2.20582.79689. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery and the formalization are the AI agent's; the agent also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the agent is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. 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Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 5 asks whether 𝒮 has an element greater than 2. This formalization decides 𝒮 versus 𝒵 for every quadratic Pisot number: the golden mean is the only one in 𝒮, so none greater than 2 lies there and Problem 5 loses its whole natural class of candidates, though the problem itself stays open. The case that had resisted the source paper's four constructions is α = 1 + √2, settled by an exact computation rather than an estimate: with ξ = (2−√2)/4 one has ⌊ξ(1+√2)^(n+1)⌋ = Pₙ − [n odd] for the Pell numbers P, so every integral part is even. The mechanism generalizes: for a real quadratic pair with second conjugate in (−1,0), the closed-form multiplier ξ = −β/(α−β) makes m divide ⌊ξαⁿ⌋ for every n ≥ 0 whenever m ∣ a and m ∣ b−1 — an exact congruence, not a confinement of fractional parts — and all but finitely many of those integral parts are then composite. The informal account is the companion paper doi:10.13140/RG.2.2.20582.79689. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery and the formalization are the AI agent's; the agent also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the agent is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. 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Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 5 asks whether 𝒮 has an element greater than 2. This formalization decides 𝒮 versus 𝒵 for every quadratic Pisot number: the golden mean is the only one in 𝒮, so none greater than 2 lies there and Problem 5 loses its whole natural class of candidates, though the problem itself stays open. The case that had resisted the source paper's four constructions is α = 1 + √2, settled by an exact computation rather than an estimate: with ξ = (2−√2)/4 one has ⌊ξ(1+√2)^(n+1)⌋ = Pₙ − [n odd] for the Pell numbers P, so every integral part is even. The mechanism generalizes: for a real quadratic pair with second conjugate in (−1,0), the closed-form multiplier ξ = −β/(α−β) makes m divide ⌊ξαⁿ⌋ for every n ≥ 0 whenever m ∣ a and m ∣ b−1 — an exact congruence, not a confinement of fractional parts — and all but finitely many of those integral parts are then composite. The informal account is the companion paper doi:10.13140/RG.2.2.20582.79689. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery and the formalization are the AI agent's; the agent also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the agent is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. 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Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 5 asks whether 𝒮 has an element greater than 2. This formalization decides 𝒮 versus 𝒵 for every quadratic Pisot number: the golden mean is the only one in 𝒮, so none greater than 2 lies there and Problem 5 loses its whole natural class of candidates, though the problem itself stays open. The case that had resisted the source paper's four constructions is α = 1 + √2, settled by an exact computation rather than an estimate: with ξ = (2−√2)/4 one has ⌊ξ(1+√2)^(n+1)⌋ = Pₙ − [n odd] for the Pell numbers P, so every integral part is even. The mechanism generalizes: for a real quadratic pair with second conjugate in (−1,0), the closed-form multiplier ξ = −β/(α−β) makes m divide ⌊ξαⁿ⌋ for every n ≥ 0 whenever m ∣ a and m ∣ b−1 — an exact congruence, not a confinement of fractional parts — and all but finitely many of those integral parts are then composite. The informal account is the companion paper doi:10.13140/RG.2.2.20582.79689. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery and the formalization are the AI agent's; the agent also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the agent is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. 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The mechanism generalizes: for a real quadratic pair with second conjugate in (−1,0), the closed-form multiplier ξ = −β/(α−β) makes m divide ⌊ξαⁿ⌋ for every n ≥ 0 whenever m ∣ a and m ∣ b−1 — an exact congruence, not a confinement of fractional parts — and all but finitely many of those integral parts are then composite. The informal account is the companion paper doi:10.13140/RG.2.2.20582.79689. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery and the formalization are the AI agent's; the agent also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the agent is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. 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The compared theorem uses a finite prime-sign cylinder and proves, uniformly in the critical window |L - log_2 N| <= C, that the total-variation distance to Pois(N/2^L) is O_C((log log N)^-2). It is conditional on three fully stated ordinary propositions specialized from Arratia--Goldstein--Gordon, Evertse--Silverman, and Nicolas--Robin; those propositions are theorem arguments, not Lean axioms.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000007", "classification": { "arxiv": [ "math.NT", "math.PR" ], "msc2020": [ "11K65", "60F05" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000008:Erdos266.erdos_266", "name": "Erdos266.erdos_266", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000008", "entry_name": "benkeene/erdos266", "repository": "https://github.com/benkeene/erdos266", "commit": "aa0cd43fb45d4f8a3019bf8925424c06b1e67874", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalization, against Mathlib, of the Kovač–Tao resolution of Erdős problem #266 (a conjecture of Stolarsky): it is not true that every sequence of positive integers with convergent reciprocal sum admits an integer shift t >= 1 making the shifted Ahmes series irrational. 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Dithered restarts keep the error below the doubly exponential schedule delta^(theta^m), for any theta in (1,2), from every starting point -- the breakdown set included -- with an explicit failure-probability exponent over the dither; in the root sense the convergence order is at least theta. Under simple spectrum, the error follows a quadratically shrinking dither schedule (root order at least 2) with failure probability linear in the initial radius up to a logarithmic factor, and MPE coincides with Reduced Rank Extrapolation at every cycle. With no dither at all, the restarted iteration from a single uniformly random starting point at worst squares the error at every cycle with high probability, under two checkable hypotheses on the Jacobian and the quadratic Taylor coefficients. The registered statements are self-contained over mathlib: every object of the construction is introduced by an explicit defining equation among the hypotheses.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000009", "classification": { "arxiv": [ "math.NA" ], "msc2020": [ "65B05", "65F10", "65H10", "41A17", "60D05" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000009:MPE.mpe_quadratic", "name": "MPE.mpe_quadratic", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000009", "entry_name": "sloisel/MPE-Formal", "repository": "https://github.com/sloisel/MPE-Formal", "commit": "149d06281b1e5f2b2af85e03f8492e1b793ab4f9", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Machine-checked proofs of three probabilistic convergence theorems for restarted Minimal Polynomial Extrapolation (MPE) applied to nonlinear fixed-point iterations. Dithered restarts keep the error below the doubly exponential schedule delta^(theta^m), for any theta in (1,2), from every starting point -- the breakdown set included -- with an explicit failure-probability exponent over the dither; in the root sense the convergence order is at least theta. Under simple spectrum, the error follows a quadratically shrinking dither schedule (root order at least 2) with failure probability linear in the initial radius up to a logarithmic factor, and MPE coincides with Reduced Rank Extrapolation at every cycle. With no dither at all, the restarted iteration from a single uniformly random starting point at worst squares the error at every cycle with high probability, under two checkable hypotheses on the Jacobian and the quadratic Taylor coefficients. The registered statements are self-contained over mathlib: every object of the construction is introduced by an explicit defining equation among the hypotheses.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000009", "classification": { "arxiv": [ "math.NA" ], "msc2020": [ "65B05", "65F10", "65H10", "41A17", "60D05" ] } }, { "id": "palomar:PALOMAR-2026-08-20-000009:MPE.mpe_undithered", "name": "MPE.mpe_undithered", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-20-000009", "entry_name": "sloisel/MPE-Formal", "repository": "https://github.com/sloisel/MPE-Formal", "commit": "149d06281b1e5f2b2af85e03f8492e1b793ab4f9", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Machine-checked proofs of three probabilistic convergence theorems for restarted Minimal Polynomial Extrapolation (MPE) applied to nonlinear fixed-point iterations. Dithered restarts keep the error below the doubly exponential schedule delta^(theta^m), for any theta in (1,2), from every starting point -- the breakdown set included -- with an explicit failure-probability exponent over the dither; in the root sense the convergence order is at least theta. Under simple spectrum, the error follows a quadratically shrinking dither schedule (root order at least 2) with failure probability linear in the initial radius up to a logarithmic factor, and MPE coincides with Reduced Rank Extrapolation at every cycle. With no dither at all, the restarted iteration from a single uniformly random starting point at worst squares the error at every cycle with high probability, under two checkable hypotheses on the Jacobian and the quadratic Taylor coefficients. The registered statements are self-contained over mathlib: every object of the construction is introduced by an explicit defining equation among the hypotheses.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-20-000009", "classification": { "arxiv": [ "math.NA" ], "msc2020": [ "65B05", "65F10", "65H10", "41A17", "60D05" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000001:WignerReal.exists_isometry_of_transProbPreserving", "name": "WignerReal.exists_isometry_of_transProbPreserving", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000001", "entry_name": "ehrlich-b/sequential-products-lean", "repository": "https://github.com/ehrlich-b/sequential-products-lean", "commit": "3c0e870eb2acb1320b62e62c791c9365f7e5260b", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A sequential product on the effects of an order-unit space is a binary operation a & b, read \"first test a, then test b with the state updated by the first test\", satisfying the seven axioms S1-S7 of van de Wetering (arXiv:1803.11139, Definition 2). This development registers two classification theorems for such products on the Hermitian matrix algebras.\nOn H_N(R), the Hermitian N x N real matrices with N >= 1, every sequential product is the Lueders product sqrt(a) b sqrt(a): there is no free parameter. On H_N(C) with N >= 3, every sequential product has the twist normal form a^(1/2+it) b a^(1/2-it) for a unique real t. The twist family is a one-parameter deformation of the Lueders product, and the complex theorem says nothing outside it satisfies the axioms. The equations are asserted on all effects, including singular ones, and the uniqueness of t is part of the registered statement.\nDirection. Both registered classification theorems are necessity: a product satisfying the axioms is supplied, and the theorem says it has the stated normal form. The two converses differ in status and neither is registered. For the complex case, that each real t defines a SeqProd satisfying S2 is proved in the wider development; together with the registered uniqueness it is what would make the parameter map injective. For the real case, that the Lueders formula defines a SeqProd satisfying S2 is not proved anywhere in this development. Nothing here registers a bijection between parameters and products, or the admissibility of the Lueders formula.\nThe package also registers two classical theorems the development proves in passing. Albert's power-associativity theorem: in a commutative Jordan ring carrying an R-module structure, x^(m+1) * x^(n+1) = x^(m+n+2), with no unit, finite dimension, formal reality or positivity assumed. And the real case of Wigner's theorem: every transition probability preserving self-map of the rays of a finite-dimensional real inner product space is induced by a linear isometry, with bijectivity not assumed but derived.\nBoth classification theorems are proved from S1-S7 with no hypotheses beyond the dimension bounds, and their axiom closures contain only Lean's three core axioms (propext, Classical.choice, Quot.sound). The continuity axiom S2 is stated in the order-unit norm, as an explicit epsilon-delta condition, so the statement file depends on no normed-space or topological instance; the equivalence with continuity in the carrier's Frobenius topology is proved in the development.\nScope. Comparator registers four theorems across three separate entries: the two sequential-product classifications above, which use the SeqProd interface and the axiom S2 and correspond to numbered rows of the manuscript; and the two ancillary classical theorems, Albert and Wigner, which use neither and correspond to no manuscript row. Nothing beyond those four registered theorems is claimed by this submission.\nThe two classification theorems are two rows of a thirty-six row manifest tracking the numbered results of the accompanying manuscript. The manuscript as a whole is not formalized: as of 2026-08-12 the manifest stands at 12 rows formalized, 19 partial, and 5 absent. In particular the quaternionic row H_n(H) and the Albert-algebra row H_3(O) of the manuscript's main theorem are not proved here, and the coalescence and diagonal-homomorphism lemmas are established on the concrete carrier H_N(C) rather than at the manuscript's stated generality of a finite-dimensional simple Euclidean Jordan algebra.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000001", "classification": { "arxiv": [ "math.OA", "math.RA", "quant-ph" ], "msc2020": [ "17C65", "81P16", "46L70", "81P10", "15B57", "17C90" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000002:BarnettePalomar.concrete_graph_certificate", "name": "BarnettePalomar.concrete_graph_certificate", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000002", "entry_name": "lennrt/palomar-formalizations", "repository": "https://github.com/lennrt/palomar-formalizations", "commit": "090b3b300ba8ec76fdba09a17f086953d13f2653", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A kernel-checked certificate for the finite mathematical core of the paper's explicit 16-vertex counterexample. 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Let w = A^(-1)1 and c* = <1, A^(-1)1>. Over the source-admissible class of scalar profiles v(s) = phi(Ls)^2 induced by even, radially nonincreasing C^2 windows supported exactly on [-L/2, L/2] with 0 <= phi <= 1 and uniform L^1 bounds on phi'' and (phi^2)'', the supremum of <1,v>^2 / equals c*, and the infimum of the reciprocal quotient / <1,v>^2 equals 1/c*. The upper bound <1,v>^2 <= c* is energy Cauchy-Schwarz. It holds for every v, it is classical, and it is not the content of this result. The content is the reverse inequality over the constrained class: c* is the value of the unconstrained variational problem, and the identity says that imposing evenness, radial monotonicity, an exact compact support, the amplitude ceiling 0 <= phi <= 1 and uniform L^1 bounds on the second derivatives does not lower the supremum. That direction is proved by exhibiting an explicit endpoint-tapered family inside the class, indexed by the scale L, whose quotient converges to c*. Both orientations are advertised because the orientation is load-bearing: it is the quotient with <1,v>^2 in the numerator whose supremum is c*, and the supporting development also proves that the two are not interchangeable. The advertised statement is existential in the profile and in the two uniform constants, so it carries its own non-vacuity: an IsLUB or IsGLB assertion over the image of an empty class is unsatisfiable in the reals, so the theorem cannot hold unless the class is populated by the constants for which it is stated. Audience. The kernel F1 is the pair-correlation form factor and the window class is the generalized, load-bearing class of Section 7.1 of the cited paper rather than a convenient flat-topped realization, so the constrained supremum settled here is the quantity that a critical-line proportion argument built on that framework optimizes; analytic number theorists working on the proportion of zeros of the Riemann zeta function on the critical line, and on pair correlation of its zeros, are the audience for which the exact value of that supremum over the admissible class, rather than a bound on it, is the relevant fact. The statement is also a closure result for a constrained admissible class in the variational theory of Fredholm integral operators (MSC 45B05, 49R05) and can be read there without reference to its origin. What the theorem itself asserts is bounded by the scope field below: it is a statement about a Fredholm operator on an interval and a class of test profiles, and it is evidence for nothing beyond that.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000004", "classification": { "arxiv": [ "math.NT", "math.CA" ], "msc2020": [ "45B05", "49R05", "11M26" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000004:ZetaLean.Palomar.pub1_strong_closure_reciprocal", "name": "ZetaLean.Palomar.pub1_strong_closure_reciprocal", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000004", "entry_name": "teal-sea/zeta-lab", "repository": "https://github.com/teal-sea/zeta-lab", "commit": "8a28a4faa3fb0fe68649f4f22faba7efe4bc3406", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A variational identity for the Fredholm operator A = I + T on the interval I = [-1/2, 1/2], where T is convolution against the Farmer-Gonek-Lee pair-correlation form factor F1(x) = |x| - 4x^2 + sum_k a_k |x|^(2k+3) with a_k = 2^(2k+3) k! / (2k+2)!. Let w = A^(-1)1 and c* = <1, A^(-1)1>. Over the source-admissible class of scalar profiles v(s) = phi(Ls)^2 induced by even, radially nonincreasing C^2 windows supported exactly on [-L/2, L/2] with 0 <= phi <= 1 and uniform L^1 bounds on phi'' and (phi^2)'', the supremum of <1,v>^2 / equals c*, and the infimum of the reciprocal quotient / <1,v>^2 equals 1/c*. The upper bound <1,v>^2 <= c* is energy Cauchy-Schwarz. It holds for every v, it is classical, and it is not the content of this result. The content is the reverse inequality over the constrained class: c* is the value of the unconstrained variational problem, and the identity says that imposing evenness, radial monotonicity, an exact compact support, the amplitude ceiling 0 <= phi <= 1 and uniform L^1 bounds on the second derivatives does not lower the supremum. That direction is proved by exhibiting an explicit endpoint-tapered family inside the class, indexed by the scale L, whose quotient converges to c*. Both orientations are advertised because the orientation is load-bearing: it is the quotient with <1,v>^2 in the numerator whose supremum is c*, and the supporting development also proves that the two are not interchangeable. The advertised statement is existential in the profile and in the two uniform constants, so it carries its own non-vacuity: an IsLUB or IsGLB assertion over the image of an empty class is unsatisfiable in the reals, so the theorem cannot hold unless the class is populated by the constants for which it is stated. Audience. The kernel F1 is the pair-correlation form factor and the window class is the generalized, load-bearing class of Section 7.1 of the cited paper rather than a convenient flat-topped realization, so the constrained supremum settled here is the quantity that a critical-line proportion argument built on that framework optimizes; analytic number theorists working on the proportion of zeros of the Riemann zeta function on the critical line, and on pair correlation of its zeros, are the audience for which the exact value of that supremum over the admissible class, rather than a bound on it, is the relevant fact. The statement is also a closure result for a constrained admissible class in the variational theory of Fredholm integral operators (MSC 45B05, 49R05) and can be read there without reference to its origin. 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Let w = A^(-1)1 and c* = <1, A^(-1)1>. Over the source-admissible class of scalar profiles v(s) = phi(Ls)^2 induced by even, radially nonincreasing C^2 windows supported exactly on [-L/2, L/2] with 0 <= phi <= 1 and uniform L^1 bounds on phi'' and (phi^2)'', the supremum of <1,v>^2 / equals c*, and the infimum of the reciprocal quotient / <1,v>^2 equals 1/c*. The upper bound <1,v>^2 <= c* is energy Cauchy-Schwarz. It holds for every v, it is classical, and it is not the content of this result. The content is the reverse inequality over the constrained class: c* is the value of the unconstrained variational problem, and the identity says that imposing evenness, radial monotonicity, an exact compact support, the amplitude ceiling 0 <= phi <= 1 and uniform L^1 bounds on the second derivatives does not lower the supremum. That direction is proved by exhibiting an explicit endpoint-tapered family inside the class, indexed by the scale L, whose quotient converges to c*. Both orientations are advertised because the orientation is load-bearing: it is the quotient with <1,v>^2 in the numerator whose supremum is c*, and the supporting development also proves that the two are not interchangeable. The advertised statement is existential in the profile and in the two uniform constants, so it carries its own non-vacuity: an IsLUB or IsGLB assertion over the image of an empty class is unsatisfiable in the reals, so the theorem cannot hold unless the class is populated by the constants for which it is stated. Audience. 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"lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A sequential product on the effects of an order-unit space is a binary operation a & b, read \"first test a, then test b with the state updated by the first test\", satisfying the seven axioms S1-S7 of van de Wetering (arXiv:1803.11139, Definition 2). This development registers two classification theorems for such products on the Hermitian matrix algebras.\nOn H_N(R), the Hermitian N x N real matrices with N >= 1, every sequential product is the Lueders product sqrt(a) b sqrt(a): there is no free parameter. On H_N(C) with N >= 3, every sequential product has the twist normal form a^(1/2+it) b a^(1/2-it) for a unique real t. The twist family is a one-parameter deformation of the Lueders product, and the complex theorem says nothing outside it satisfies the axioms.\nPrior art. That Lueders is not the only sequential product, and twisted products of this shape in particular, are not new: Gudder asked the question in 2005, Shen and Wu answered it negatively and gave a general construction in 2009 (arXiv:0905.0596), and Liu, Wu and Wu showed those examples are unitary equivalent to Lueders (arXiv:1009.2593). What is registered here is the converse direction. The examples are not exhibited; they are shown to be forced, with the parameter unique, and with no product outside the family admissible. The equations are asserted on all effects, including singular ones, and the uniqueness of t is part of the registered statement.\nDirection. Both registered classification theorems are necessity: a product satisfying the axioms is supplied, and the theorem says it has the stated normal form. The two converses differ in status and neither is registered. For the complex case, that each real t defines a SeqProd satisfying S2 is proved in the wider development; together with the registered uniqueness it is what would make the parameter map injective. For the real case, that the Lueders formula defines a SeqProd satisfying S2 is not proved anywhere in this development. Nothing here registers a bijection between parameters and products, or the admissibility of the Lueders formula.\nThe package also registers two classical theorems the development proves in passing. Albert's power-associativity theorem: in a commutative Jordan ring carrying an R-module structure, x^(m+1) * x^(n+1) = x^(m+n+2), with no unit, finite dimension, formal reality or positivity assumed. And the real case of Wigner's theorem: every transition probability preserving self-map of the rays of a finite-dimensional real inner product space is induced by a linear isometry, with bijectivity not assumed but derived.\nBoth classification theorems are proved from S1-S7 with no hypotheses beyond the dimension bounds, and their axiom closures contain only Lean's three core axioms (propext, Classical.choice, Quot.sound). The continuity axiom S2 is stated in the order-unit norm, as an explicit epsilon-delta condition, so the statement file depends on no normed-space or topological instance; the equivalence with continuity in the carrier's Frobenius topology is proved in the development.\nScope. The repository carries three Comparator configurations, selecting four theorems between them: the two sequential-product classifications above, which use the SeqProd interface and the axiom S2 and correspond to numbered rows of the manuscript; and the two ancillary classical theorems, Albert and Wigner, which use neither and correspond to no manuscript row. Each configuration is a separate submission and a separate entry; no one entry contains all four. No Palomar claim is made here for any other declaration in the repository.\nThe two classification theorems are two rows of a thirty-six row manifest tracking the numbered results of the accompanying manuscript. The manuscript as a whole is not formalized: as of 2026-08-12 the manifest stands at 12 rows formalized, 19 partial, and 5 absent. In particular the quaternionic row H_n(H) and the Albert-algebra row H_3(O) of the manuscript's main theorem are not proved here, and the coalescence and diagonal-homomorphism lemmas are established on the concrete carrier H_N(C) rather than at the manuscript's stated generality of a finite-dimensional simple Euclidean Jordan algebra.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000009", "classification": { "arxiv": [ "math.OA", "math.RA", "quant-ph" ], "msc2020": [ "17C65", "81P16", "46L70", "81P10", "15B57", "17C90" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000010:PalomarCubicHalfSpectrum.released_outer_source_normal_form", "name": "PalomarCubicHalfSpectrum.released_outer_source_normal_form", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000010", "entry_name": "lennrt/palomar-formalizations", "repository": "https://github.com/lennrt/palomar-formalizations", "commit": "d259d35ad32d0e7c8c60ddd5a574aa987956e1d2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean formalization of Lemma 7.1 and the outer-matrix content of Theorem 7.2 for generalized cubic subdivision. It defines the naturally indexed 28n by 28n outer matrix row-by-row from the authenticated released MATLAB masks, including all previous/current/next-sector references. It constructs the paper's explicit topological permutation, proves that the first 24n coordinates are strictly upper triangular and that the remaining diagonal components are n scalar 1/64 blocks and n copies of the displayed 3 by 3 core, and proves unconditionally for every n at least 3 the characteristic polynomial X^(24n)(X-1/64)^n(X-1/32)^n(X-1/16)^n(X-1/8)^n.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000010", "classification": { "arxiv": [ "cs.GR", "math.NA" ], "msc2020": [ "15A18", "65D07", "65D17", "65F15" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000010:PalomarCubicHalfSpectrum.released_zero_block_strictly_upper", "name": "PalomarCubicHalfSpectrum.released_zero_block_strictly_upper", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000010", "entry_name": "lennrt/palomar-formalizations", "repository": "https://github.com/lennrt/palomar-formalizations", "commit": "d259d35ad32d0e7c8c60ddd5a574aa987956e1d2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean formalization of Lemma 7.1 and the outer-matrix content of Theorem 7.2 for generalized cubic subdivision. It defines the naturally indexed 28n by 28n outer matrix row-by-row from the authenticated released MATLAB masks, including all previous/current/next-sector references. It constructs the paper's explicit topological permutation, proves that the first 24n coordinates are strictly upper triangular and that the remaining diagonal components are n scalar 1/64 blocks and n copies of the displayed 3 by 3 core, and proves unconditionally for every n at least 3 the characteristic polynomial X^(24n)(X-1/64)^n(X-1/32)^n(X-1/16)^n(X-1/8)^n.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000010", "classification": { "arxiv": [ "cs.GR", "math.NA" ], "msc2020": [ "15A18", "65D07", "65D17", "65F15" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000010:PalomarCubicHalfSpectrum.outer_core_charpoly", "name": "PalomarCubicHalfSpectrum.outer_core_charpoly", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000010", "entry_name": "lennrt/palomar-formalizations", "repository": "https://github.com/lennrt/palomar-formalizations", "commit": "d259d35ad32d0e7c8c60ddd5a574aa987956e1d2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean formalization of Lemma 7.1 and the outer-matrix content of Theorem 7.2 for generalized cubic subdivision. It defines the naturally indexed 28n by 28n outer matrix row-by-row from the authenticated released MATLAB masks, including all previous/current/next-sector references. It constructs the paper's explicit topological permutation, proves that the first 24n coordinates are strictly upper triangular and that the remaining diagonal components are n scalar 1/64 blocks and n copies of the displayed 3 by 3 core, and proves unconditionally for every n at least 3 the characteristic polynomial X^(24n)(X-1/64)^n(X-1/32)^n(X-1/16)^n(X-1/8)^n.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000010", "classification": { "arxiv": [ "cs.GR", "math.NA" ], "msc2020": [ "15A18", "65D07", "65D17", "65F15" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000010:PalomarCubicHalfSpectrum.outer_normal_form_charpoly", "name": "PalomarCubicHalfSpectrum.outer_normal_form_charpoly", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000010", "entry_name": "lennrt/palomar-formalizations", "repository": "https://github.com/lennrt/palomar-formalizations", "commit": "d259d35ad32d0e7c8c60ddd5a574aa987956e1d2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean formalization of Lemma 7.1 and the outer-matrix content of Theorem 7.2 for generalized cubic subdivision. It defines the naturally indexed 28n by 28n outer matrix row-by-row from the authenticated released MATLAB masks, including all previous/current/next-sector references. It constructs the paper's explicit topological permutation, proves that the first 24n coordinates are strictly upper triangular and that the remaining diagonal components are n scalar 1/64 blocks and n copies of the displayed 3 by 3 core, and proves unconditionally for every n at least 3 the characteristic polynomial X^(24n)(X-1/64)^n(X-1/32)^n(X-1/16)^n(X-1/8)^n.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000010", "classification": { "arxiv": [ "cs.GR", "math.NA" ], "msc2020": [ "15A18", "65D07", "65D17", "65F15" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000010:PalomarCubicHalfSpectrum.released_outer_charpoly", "name": "PalomarCubicHalfSpectrum.released_outer_charpoly", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000010", "entry_name": "lennrt/palomar-formalizations", "repository": "https://github.com/lennrt/palomar-formalizations", "commit": "d259d35ad32d0e7c8c60ddd5a574aa987956e1d2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean formalization of Lemma 7.1 and the outer-matrix content of Theorem 7.2 for generalized cubic subdivision. It defines the naturally indexed 28n by 28n outer matrix row-by-row from the authenticated released MATLAB masks, including all previous/current/next-sector references. It constructs the paper's explicit topological permutation, proves that the first 24n coordinates are strictly upper triangular and that the remaining diagonal components are n scalar 1/64 blocks and n copies of the displayed 3 by 3 core, and proves unconditionally for every n at least 3 the characteristic polynomial X^(24n)(X-1/64)^n(X-1/32)^n(X-1/16)^n(X-1/8)^n.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000010", "classification": { "arxiv": [ "cs.GR", "math.NA" ], "msc2020": [ "15A18", "65D07", "65D17", "65F15" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000011:PalomarMSET25.cycle_distance_pair_locus", "name": "PalomarMSET25.cycle_distance_pair_locus", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000011", "entry_name": "lennrt/palomar-formalizations", "repository": "https://github.com/lennrt/palomar-formalizations", "commit": "d259d35ad32d0e7c8c60ddd5a574aa987956e1d2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A concrete formalization of the infinite-family construction in the source: cycle and Cartesian-product distances on P_m □ C_(6m), the displayed landmarks (0,0), (0,2m), and (m-1,0), the four-piece cycle distance-pair locus, and injectivity of the resulting unordered multiset code for every even m at least 2. This establishes the three-landmark resolving upper bound. It does not formalize the separate general lower bound needed to conclude that the set is a basis or that the multiset dimension equals 3.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000011", "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C12", "05C76", "68R10" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000011:PalomarMSET25.cycle_distance_pair_injective", "name": "PalomarMSET25.cycle_distance_pair_injective", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000011", "entry_name": "lennrt/palomar-formalizations", "repository": "https://github.com/lennrt/palomar-formalizations", "commit": "d259d35ad32d0e7c8c60ddd5a574aa987956e1d2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A concrete formalization of the infinite-family construction in the source: cycle and Cartesian-product distances on P_m □ C_(6m), the displayed landmarks (0,0), (0,2m), and (m-1,0), the four-piece cycle distance-pair locus, and injectivity of the resulting unordered multiset code for every even m at least 2. This establishes the three-landmark resolving upper bound. It does not formalize the separate general lower bound needed to conclude that the set is a basis or that the multiset dimension equals 3.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000011", "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C12", "05C76", "68R10" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000011:PalomarMSET25.even_cylinder_three_landmarks_resolve", "name": "PalomarMSET25.even_cylinder_three_landmarks_resolve", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000011", "entry_name": "lennrt/palomar-formalizations", "repository": "https://github.com/lennrt/palomar-formalizations", "commit": "d259d35ad32d0e7c8c60ddd5a574aa987956e1d2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A concrete formalization of the infinite-family construction in the source: cycle and Cartesian-product distances on P_m □ C_(6m), the displayed landmarks (0,0), (0,2m), and (m-1,0), the four-piece cycle distance-pair locus, and injectivity of the resulting unordered multiset code for every even m at least 2. This establishes the three-landmark resolving upper bound. It does not formalize the separate general lower bound needed to conclude that the set is a basis or that the multiset dimension equals 3.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000011", "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C12", "05C76", "68R10" ] } }, { "id": "palomar:PALOMAR-2026-08-21-000012:ZetaLean.PalomarDH.dh_analytic_half", "name": "ZetaLean.PalomarDH.dh_analytic_half", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-21-000012", "entry_name": "teal-sea/zeta-lab", "repository": "https://github.com/teal-sea/zeta-lab", "commit": "097215a8413641fd3b3de137386316bf294328de", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Davenport and Heilbronn (1936) exhibited a Dirichlet series with real coefficients satisfying a Riemann-type functional equation which nevertheless has zeros off the critical line. It is the standard demonstration that a functional equation of Riemann type does not on its own force the Riemann Hypothesis, it is the reference counterexample against which structural explanations of the Riemann Hypothesis are tested, and the location of its zeros remains an active subject. The function is DH(s) = (1 - i*kappa)/2 * L(s, chi) + (1 + i*kappa)/2 * L(s, chi^-1) for a quartic Dirichlet character chi mod 5 and kappa = (sqrt(10 - 2*sqrt 5) - 2)/(sqrt 5 - 1), the value that rotates the two conjugate root numbers onto each other and so makes the Dirichlet coefficients real. This submission advertises one theorem, the analytic half of the Davenport-Heilbronn theorem: there exists an entire function represented by the Davenport-Heilbronn Dirichlet series on Re z > 1 whose completion (pi/5)^(-(s+1)/2) * Gamma((s+1)/2) * f(s) is symmetric under s -> 1-s. The statement is existential in f, so it carries its own non-vacuity and needs none of the character theory used to construct the witness. What the work consists of. The Davenport-Heilbronn function is not a library object anywhere we could find, so the development builds it: the quartic character mod 5, the linear combination, entirety on all of C, the identification of the Dirichlet coefficients as the real period-5 sequence 1, kappa, -kappa, -1, 0, and the completed functional equation. The last of these is where the content sits. It reduces to a root-number identity whose convention-sensitive constants are derived inside the development, from Mathlib's Real.cos_pi_div_five by radical algebra, rather than transcribed from a table; the standard sources state kappa without deriving it in a form a kernel will accept, and getting it wrong yields a function whose coefficients are not real and which therefore is not the counterexample. The functional equation is stated with explicit guards excluding the poles of the two Gamma factors, because Mathlib's junk value Gamma = 0 at the poles would otherwise make the unrestricted equation false for reasons having nothing to do with the mathematics. Audience. The Davenport-Heilbronn function is a named object of analytic number theory with a continuing literature on its zeros, so a machine-checked construction of it is of direct interest to number theorists working on zeros of L-functions and on what a Riemann-type functional equation does and does not imply. It is also the object a formalized treatment of the Davenport-Heilbronn theorem has to have before it can say anything, and this supplies it in a form that carries its own non-vacuity. Prior art. A search of Mathlib at the revision pinned by this repository found no Davenport-Heilbronn function and no formalization of this functional equation. That is a bounded search of one library. No systematic search of other proof assistants or of the mathematical literature for prior formalizations has been run, and no novelty against the literature is claimed: the mathematics is Davenport and Heilbronn's, published in 1936. Scope. The Davenport-Heilbronn theorem itself is NOT proved here. Its full statement additionally requires a zero s with Re s different from 1/2, and that conjunct appears nowhere in the advertised theorem. What is offered is the analytic half alone. No numerical enclosure for any zero of DH is asserted, and nothing here is a statement about the Riemann zeta function or the Riemann Hypothesis.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-21-000012", "classification": { "arxiv": [ "math.NT", "math.CV" ], "msc2020": [ "11M26", "11M41" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000001:Palomar.Erdos730.erdos_730_infinite", "name": "Palomar.Erdos730.erdos_730_infinite", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000001", "entry_name": "williamjblair/lean-proofs", "repository": "https://github.com/williamjblair/lean-proofs", "commit": "03729c9cbb0b602f5a828bb850c85e84c5a6d460", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Machine-checked Lean 4 proofs of solved research problems from the Erdős problems collection, hosted as stable formal_proof targets for Google DeepMind's Formal Conjectures. The principal result is an affirmative answer to Erdős Problem #730 (Erdős, Graham, Ruzsa and Straus, 1975): the set of pairs n < m for which binom(2n, n) and binom(2m, m) have the same set of prime divisors is infinite, and in fact contains infinitely many consecutive pairs (n, n + 1). The proof formalises the argument posted informally by Liam Price on the problem's discussion page on 24 June 2026: Kummer's theorem converts equality of prime supports into a base-p digit condition, an explicit quadratic family of consecutive pairs is constructed, a fixed-depth Fourier estimate bounds the digit obstruction on blocks of length p^r, a uniform depth tail uses a Mertens-type reciprocal-prime input, the top-prime range is handled by fixed-modulus divisor switching using the prime number theorem in arithmetic progressions, and an exact rational density budget yields positive lower density and hence infinitude. The repository also proves the sumset form of Erdős Problem #154 (equidistribution of A + A in residue classes for Sidon sets A with |A| ~ sqrt N, from Lindström's theorem) and hosts smaller bounded results for Problems #94, #399 and #1074. Every tracked theorem depends only on propext, Classical.choice and Quot.sound; each registry statement under Palomar/ restates its result in a Challenge importing Mathlib alone.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000001", "classification": { "arxiv": [ "math.NT", "math.CO" ], "msc2020": [ "11B65", "11N13", "11A41", "11B13" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000002:RiemannHypothesisProofFactory.SelbergConditioning.near_linear_conditioning_witness", "name": "RiemannHypothesisProofFactory.SelbergConditioning.near_linear_conditioning_witness", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000002", "entry_name": "mbaccaro-dev/mathematical-proofs", "repository": "https://github.com/mbaccaro-dev/mathematical-proofs", "commit": "f0ccae19f2cb5afdca00c42694d677c74dc492f6", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 proof of the sequence-level conditioning construction for Selberg divisor feedback. Two explicit perturbations of the von Mangoldt weight stay in the positive prime-power cone, approximate the divisor data to an arbitrary tolerance through twice the observation scale, separate their summatory outputs at a near-linear power scale, and retain one uniform linear Selberg bound.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000002", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N05", "11M26" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000003:PalomarHahnHeckeAppell.kronecker_partial_fraction_cleared_coefficient", "name": "PalomarHahnHeckeAppell.kronecker_partial_fraction_cleared_coefficient", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000003", "entry_name": "xiangyazi24/palomar-hahn-hecke-appell", "repository": "https://github.com/xiangyazi24/palomar-hahn-hecke-appell", "commit": "b9d84fd9f905605b3956f974f50ed41ee59d54e2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 verification of the coefficientwise Kronecker partial-fraction identity and the general boundary-cleared Hickerson--Mortenson Hecke--Appell identity for f_{n,n+p,n}, for all positive integers n and p.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000003", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11F27", "33D15" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000003:PalomarHahnHeckeAppell.general_boundary_cleared_hm_coefficient", "name": "PalomarHahnHeckeAppell.general_boundary_cleared_hm_coefficient", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000003", "entry_name": "xiangyazi24/palomar-hahn-hecke-appell", "repository": "https://github.com/xiangyazi24/palomar-hahn-hecke-appell", "commit": "b9d84fd9f905605b3956f974f50ed41ee59d54e2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 verification of the coefficientwise Kronecker partial-fraction identity and the general boundary-cleared Hickerson--Mortenson Hecke--Appell identity for f_{n,n+p,n}, for all positive integers n and p.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000003", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11F27", "33D15" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000004:RiemannHypothesisProofFactory.SubpowerDyadicEnergy.dyadic_global_energy_balance", "name": "RiemannHypothesisProofFactory.SubpowerDyadicEnergy.dyadic_global_energy_balance", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000004", "entry_name": "mbaccaro-dev/mathematical-proofs", "repository": "https://github.com/mbaccaro-dev/mathematical-proofs", "commit": "f0ccae19f2cb5afdca00c42694d677c74dc492f6", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 proof of the global power-of-two balance connecting dyadic defect, block variation, and positive Möbius-logarithm prefix energy. Eventual nonnegativity on exactly those dyadic blocks gives an explicit all-endpoint cubic energy bound. With the stated weighted-squarefree remainder estimate, the formalization also recovers the sharp 6/pi^2 leading coefficient and retains the complete finite-prefix correction.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000004", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11N37" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000004:RiemannHypothesisProofFactory.SubpowerDyadicEnergy.eventualDyadicDefectFrom_implies_sharp_power_energy_bound", "name": "RiemannHypothesisProofFactory.SubpowerDyadicEnergy.eventualDyadicDefectFrom_implies_sharp_power_energy_bound", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000004", "entry_name": "mbaccaro-dev/mathematical-proofs", "repository": "https://github.com/mbaccaro-dev/mathematical-proofs", "commit": "f0ccae19f2cb5afdca00c42694d677c74dc492f6", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 proof of the global power-of-two balance connecting dyadic defect, block variation, and positive Möbius-logarithm prefix energy. Eventual nonnegativity on exactly those dyadic blocks gives an explicit all-endpoint cubic energy bound. With the stated weighted-squarefree remainder estimate, the formalization also recovers the sharp 6/pi^2 leading coefficient and retains the complete finite-prefix correction.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000004", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11N37" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000004:RiemannHypothesisProofFactory.SubpowerDyadicEnergy.eventualDyadicDefectFrom_implies_cumulative_energy_cubic_bound", "name": "RiemannHypothesisProofFactory.SubpowerDyadicEnergy.eventualDyadicDefectFrom_implies_cumulative_energy_cubic_bound", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000004", "entry_name": "mbaccaro-dev/mathematical-proofs", "repository": "https://github.com/mbaccaro-dev/mathematical-proofs", "commit": "f0ccae19f2cb5afdca00c42694d677c74dc492f6", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 proof of the global power-of-two balance connecting dyadic defect, block variation, and positive Möbius-logarithm prefix energy. Eventual nonnegativity on exactly those dyadic blocks gives an explicit all-endpoint cubic energy bound. With the stated weighted-squarefree remainder estimate, the formalization also recovers the sharp 6/pi^2 leading coefficient and retains the complete finite-prefix correction.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000004", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11M26", "11N37" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000005:RiemannHypothesisProofFactory.MobiusResidualComplexity.chebyshev_globalMatching_residual_budget_barrier", "name": "RiemannHypothesisProofFactory.MobiusResidualComplexity.chebyshev_globalMatching_residual_budget_barrier", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000005", "entry_name": "mbaccaro-dev/mathematical-proofs", "repository": "https://github.com/mbaccaro-dev/mathematical-proofs", "commit": "11d59ae23fb0fe3d8591a00069bec47a868ac93f", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 proof of a residual-or-unmatched barrier for local Möbius cancellation. At every sufficiently large fixed-band scale, either the permitted edge length reaches the retained-prime product or every member of an explicit squarefree prime-product family is unmatched. Chebyshev estimates give the exact binomial and factorial population bounds. The product scale also converts residual rank to its logarithmic threshold with the exact limiting constant.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000005", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N37", "11A25" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000005:RiemannHypothesisProofFactory.MobiusResidualComplexity.eventually_octuple_globalMatching_residual_or_unmatched_barrier", "name": "RiemannHypothesisProofFactory.MobiusResidualComplexity.eventually_octuple_globalMatching_residual_or_unmatched_barrier", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000005", "entry_name": "mbaccaro-dev/mathematical-proofs", "repository": "https://github.com/mbaccaro-dev/mathematical-proofs", "commit": "11d59ae23fb0fe3d8591a00069bec47a868ac93f", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 proof of a residual-or-unmatched barrier for local Möbius cancellation. At every sufficiently large fixed-band scale, either the permitted edge length reaches the retained-prime product or every member of an explicit squarefree prime-product family is unmatched. Chebyshev estimates give the exact binomial and factorial population bounds. The product scale also converts residual rank to its logarithmic threshold with the exact limiting constant.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000005", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N37", "11A25" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000005:RiemannHypothesisProofFactory.MobiusResidualComplexity.threshold_logarithmic_scale_conversion", "name": "RiemannHypothesisProofFactory.MobiusResidualComplexity.threshold_logarithmic_scale_conversion", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000005", "entry_name": "mbaccaro-dev/mathematical-proofs", "repository": "https://github.com/mbaccaro-dev/mathematical-proofs", "commit": "11d59ae23fb0fe3d8591a00069bec47a868ac93f", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 proof of a residual-or-unmatched barrier for local Möbius cancellation. At every sufficiently large fixed-band scale, either the permitted edge length reaches the retained-prime product or every member of an explicit squarefree prime-product family is unmatched. Chebyshev estimates give the exact binomial and factorial population bounds. The product scale also converts residual rank to its logarithmic threshold with the exact limiting constant.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000005", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N37", "11A25" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000005:RiemannHypothesisProofFactory.MobiusResidualComplexity.eventually_threshold_budget_separation", "name": "RiemannHypothesisProofFactory.MobiusResidualComplexity.eventually_threshold_budget_separation", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000005", "entry_name": "mbaccaro-dev/mathematical-proofs", "repository": "https://github.com/mbaccaro-dev/mathematical-proofs", "commit": "11d59ae23fb0fe3d8591a00069bec47a868ac93f", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 proof of a residual-or-unmatched barrier for local Möbius cancellation. At every sufficiently large fixed-band scale, either the permitted edge length reaches the retained-prime product or every member of an explicit squarefree prime-product family is unmatched. Chebyshev estimates give the exact binomial and factorial population bounds. The product scale also converts residual rank to its logarithmic threshold with the exact limiting constant.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000005", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11N37", "11A25" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000006:TwistNormalForm.real_classification", "name": "TwistNormalForm.real_classification", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000006", "entry_name": "ehrlich-b/sequential-products-lean", "repository": "https://github.com/ehrlich-b/sequential-products-lean", "commit": "9c6d92f5fc222d9d1646ae6044dcec053900ca83", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A sequential product on the effects of an order-unit space is a binary operation a & b, read \"first test a, then test b with the state updated by the first test\", satisfying the seven axioms S1-S7 of van de Wetering (arXiv:1803.11139, Definition 2). This development registers two classification theorems for such products on the Hermitian matrix algebras.\nOn H_N(R), the Hermitian N x N real matrices with N >= 1, every sequential product is the Lueders product sqrt(a) b sqrt(a): there is no free parameter. On H_N(C) with N >= 3, every sequential product has the twist normal form a^(1/2+it) b a^(1/2-it) for a unique real t. The twist family is a one-parameter deformation of the Lueders product, and the complex theorem says nothing outside it satisfies the axioms.\nPrior art. That Lueders is not the only sequential product, and twisted products of this shape in particular, are not new: Gudder asked the question in 2005, Shen and Wu answered it negatively and gave a general construction in 2009 (arXiv:0905.0596), and Liu, Wu and Wu showed those examples are unitary equivalent to Lueders (arXiv:1009.2593). What is registered here is the converse direction. The examples are not exhibited; they are shown to be forced, with the parameter unique, and with no product outside the family admissible. The equations are asserted on all effects, including singular ones, and the uniqueness of t is part of the registered statement.\nDirection. Both registered classification theorems are necessity: a product satisfying the axioms is supplied, and the theorem says it has the stated normal form. The two converses differ in status and neither is registered. For the complex case, that each real t defines a SeqProd satisfying S2 is proved in the wider development; together with the registered uniqueness it is what would make the parameter map injective. For the real case, that the Lueders formula defines a SeqProd satisfying S2 is not proved anywhere in this development. Nothing here registers a bijection between parameters and products, or the admissibility of the Lueders formula.\nThe package also registers two classical theorems the development proves in passing. Albert's power-associativity theorem: in a commutative Jordan ring carrying an R-module structure, x^(m+1) * x^(n+1) = x^(m+n+2), with no unit, finite dimension, formal reality or positivity assumed. And the real case of Wigner's theorem: every transition probability preserving self-map of the rays of a finite-dimensional real inner product space is induced by a linear isometry, with bijectivity not assumed but derived.\nBoth classification theorems are proved from S1-S7 with no hypotheses beyond the dimension bounds, and their axiom closures contain only Lean's three core axioms (propext, Classical.choice, Quot.sound). The continuity axiom S2 is stated in the order-unit norm, as an explicit epsilon-delta condition, so the statement file depends on no normed-space or topological instance; the equivalence with continuity in the carrier's Frobenius topology is proved in the development.\nScope. The repository carries three Comparator configurations, selecting four theorems between them: the two sequential-product classifications above, which use the SeqProd interface and the axiom S2 and correspond to numbered rows of the manuscript; and the two ancillary classical theorems, Albert and Wigner, which use neither and correspond to no manuscript row. Each configuration is a separate submission and a separate entry; no one entry contains all four. No Palomar claim is made here for any other declaration in the repository.\nThe two classification theorems are two rows of a thirty-six row manifest tracking the numbered results of the accompanying manuscript. The manuscript as a whole is not formalized: as of 2026-08-12 the manifest stands at 12 rows formalized, 19 partial, and 5 absent. In particular the quaternionic row H_n(H) and the Albert-algebra row H_3(O) of the manuscript's main theorem are not proved here, and the coalescence and diagonal-homomorphism lemmas are established on the concrete carrier H_N(C) rather than at the manuscript's stated generality of a finite-dimensional simple Euclidean Jordan algebra.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000006", "classification": { "arxiv": [ "math.OA", "math.RA", "quant-ph" ], "msc2020": [ "17C65", "81P16", "46L70", "81P10", "15B57", "17C90" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000006:TwistNormalForm.complex_classification", "name": "TwistNormalForm.complex_classification", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000006", "entry_name": "ehrlich-b/sequential-products-lean", "repository": "https://github.com/ehrlich-b/sequential-products-lean", "commit": "9c6d92f5fc222d9d1646ae6044dcec053900ca83", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A sequential product on the effects of an order-unit space is a binary operation a & b, read \"first test a, then test b with the state updated by the first test\", satisfying the seven axioms S1-S7 of van de Wetering (arXiv:1803.11139, Definition 2). This development registers two classification theorems for such products on the Hermitian matrix algebras.\nOn H_N(R), the Hermitian N x N real matrices with N >= 1, every sequential product is the Lueders product sqrt(a) b sqrt(a): there is no free parameter. On H_N(C) with N >= 3, every sequential product has the twist normal form a^(1/2+it) b a^(1/2-it) for a unique real t. The twist family is a one-parameter deformation of the Lueders product, and the complex theorem says nothing outside it satisfies the axioms.\nPrior art. That Lueders is not the only sequential product, and twisted products of this shape in particular, are not new: Gudder asked the question in 2005, Shen and Wu answered it negatively and gave a general construction in 2009 (arXiv:0905.0596), and Liu, Wu and Wu showed those examples are unitary equivalent to Lueders (arXiv:1009.2593). What is registered here is the converse direction. The examples are not exhibited; they are shown to be forced, with the parameter unique, and with no product outside the family admissible. The equations are asserted on all effects, including singular ones, and the uniqueness of t is part of the registered statement.\nDirection. Both registered classification theorems are necessity: a product satisfying the axioms is supplied, and the theorem says it has the stated normal form. The two converses differ in status and neither is registered. For the complex case, that each real t defines a SeqProd satisfying S2 is proved in the wider development; together with the registered uniqueness it is what would make the parameter map injective. For the real case, that the Lueders formula defines a SeqProd satisfying S2 is not proved anywhere in this development. Nothing here registers a bijection between parameters and products, or the admissibility of the Lueders formula.\nThe package also registers two classical theorems the development proves in passing. Albert's power-associativity theorem: in a commutative Jordan ring carrying an R-module structure, x^(m+1) * x^(n+1) = x^(m+n+2), with no unit, finite dimension, formal reality or positivity assumed. And the real case of Wigner's theorem: every transition probability preserving self-map of the rays of a finite-dimensional real inner product space is induced by a linear isometry, with bijectivity not assumed but derived.\nBoth classification theorems are proved from S1-S7 with no hypotheses beyond the dimension bounds, and their axiom closures contain only Lean's three core axioms (propext, Classical.choice, Quot.sound). The continuity axiom S2 is stated in the order-unit norm, as an explicit epsilon-delta condition, so the statement file depends on no normed-space or topological instance; the equivalence with continuity in the carrier's Frobenius topology is proved in the development.\nScope. The repository carries three Comparator configurations, selecting four theorems between them: the two sequential-product classifications above, which use the SeqProd interface and the axiom S2 and correspond to numbered rows of the manuscript; and the two ancillary classical theorems, Albert and Wigner, which use neither and correspond to no manuscript row. Each configuration is a separate submission and a separate entry; no one entry contains all four. No Palomar claim is made here for any other declaration in the repository.\nThe two classification theorems are two rows of a thirty-six row manifest tracking the numbered results of the accompanying manuscript. The manuscript as a whole is not formalized: as of 2026-08-12 the manifest stands at 12 rows formalized, 19 partial, and 5 absent. In particular the quaternionic row H_n(H) and the Albert-algebra row H_3(O) of the manuscript's main theorem are not proved here, and the coalescence and diagonal-homomorphism lemmas are established on the concrete carrier H_N(C) rather than at the manuscript's stated generality of a finite-dimensional simple Euclidean Jordan algebra.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000006", "classification": { "arxiv": [ "math.OA", "math.RA", "quant-ph" ], "msc2020": [ "17C65", "81P16", "46L70", "81P10", "15B57", "17C90" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000007:PalomarKinnim.physicalApportionmentExactMinimum", "name": "PalomarKinnim.physicalApportionmentExactMinimum", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000007", "entry_name": "ashebson/kinin-lean", "repository": "https://github.com/ashebson/kinin-lean", "commit": "d8ca202e6ccd655bd9c49dad5b9ee724cd3c2f54", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalization of all sixteen units of Mishnah Kinnim, with a machine-checked physical optimization theorem for the Koppel--Reiss reading of Kinnim 3:2. For any list of pair counts grouped by owner, the priest has already acted on exactly half of the mixed birds above and half below. A physical world records one action level for every individual bird of every owner. Lean proves that every such world is compatible, by permutation, with the fixed half-split action list. Counting valid birds directly from each owner's closed-pair quota, the theorem proves that the exact minimum over all physical apportionments is twice the complement of the largest attainable subset sum not exceeding half the total, and constructs an apportionment attaining it. The repository also covers all sixteen units, flight recurrences, and replacement-inventory covers.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000007", "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05A17", "90C27" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000008:KaltonPeck.rankParityGeneral", "name": "KaltonPeck.rankParityGeneral", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000008", "entry_name": "adas1236/KaltonPeckPalomar", "repository": "https://github.com/adas1236/KaltonPeckPalomar", "commit": "6830235450adb5f7da6bc683f7ac6b1767320cc0", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "We prove the following mod-2 identity for real symplectic Banach spaces:\n If $(X, \\omega)$ is a real symplectic Banach space, and $T \\in L(X)$ satisfies:\n (1) $I + T^+T$ is Fredholm\n (2) $I + T^2$ has finite rank\n Then\n $\\rank(T^2 + I) \\equiv \\dim \\ker (I + T^+T) \\pmod 2$.\nIn the above, $T^+$ represents the symplectic adjoint. See our paper for more depth in definitions. This identity is applied to show that no hyperplane of the real Kalton-Peck space has a complex structure. We note that in our notation, a symplectic Banach space always assumes that the symplectic form \\omega is a strong symplectic form. See our paper for more detail on the definitions.\nWe formalize:\n (1) The mod-2 identity for arbitrary real symplectic Banach spaces\n (2) An application of the identity to show that if $T \\in L(Z_2)$ and $T^2 + I$ has finite rank, then $T^2 + I$ has even rank.\n Here, Z_2 is the real Kalton-Peck space.\n (3) The corollary that no hyperplane of the real Kalton-Peck space has a complex structure.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000008", "classification": { "arxiv": [ "math.FA" ], "msc2020": [ "46B03", "46B20", "47A53" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000008:KaltonPeck.rankParityZ2", "name": "KaltonPeck.rankParityZ2", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000008", "entry_name": "adas1236/KaltonPeckPalomar", "repository": "https://github.com/adas1236/KaltonPeckPalomar", "commit": "6830235450adb5f7da6bc683f7ac6b1767320cc0", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "We prove the following mod-2 identity for real symplectic Banach spaces:\n If $(X, \\omega)$ is a real symplectic Banach space, and $T \\in L(X)$ satisfies:\n (1) $I + T^+T$ is Fredholm\n (2) $I + T^2$ has finite rank\n Then\n $\\rank(T^2 + I) \\equiv \\dim \\ker (I + T^+T) \\pmod 2$.\nIn the above, $T^+$ represents the symplectic adjoint. See our paper for more depth in definitions. This identity is applied to show that no hyperplane of the real Kalton-Peck space has a complex structure. We note that in our notation, a symplectic Banach space always assumes that the symplectic form \\omega is a strong symplectic form. See our paper for more detail on the definitions.\nWe formalize:\n (1) The mod-2 identity for arbitrary real symplectic Banach spaces\n (2) An application of the identity to show that if $T \\in L(Z_2)$ and $T^2 + I$ has finite rank, then $T^2 + I$ has even rank.\n Here, Z_2 is the real Kalton-Peck space.\n (3) The corollary that no hyperplane of the real Kalton-Peck space has a complex structure.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000008", "classification": { "arxiv": [ "math.FA" ], "msc2020": [ "46B03", "46B20", "47A53" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000008:KaltonPeck.noHyperplaneComplexStructure", "name": "KaltonPeck.noHyperplaneComplexStructure", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000008", "entry_name": "adas1236/KaltonPeckPalomar", "repository": "https://github.com/adas1236/KaltonPeckPalomar", "commit": "6830235450adb5f7da6bc683f7ac6b1767320cc0", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "We prove the following mod-2 identity for real symplectic Banach spaces:\n If $(X, \\omega)$ is a real symplectic Banach space, and $T \\in L(X)$ satisfies:\n (1) $I + T^+T$ is Fredholm\n (2) $I + T^2$ has finite rank\n Then\n $\\rank(T^2 + I) \\equiv \\dim \\ker (I + T^+T) \\pmod 2$.\nIn the above, $T^+$ represents the symplectic adjoint. See our paper for more depth in definitions. This identity is applied to show that no hyperplane of the real Kalton-Peck space has a complex structure. We note that in our notation, a symplectic Banach space always assumes that the symplectic form \\omega is a strong symplectic form. See our paper for more detail on the definitions.\nWe formalize:\n (1) The mod-2 identity for arbitrary real symplectic Banach spaces\n (2) An application of the identity to show that if $T \\in L(Z_2)$ and $T^2 + I$ has finite rank, then $T^2 + I$ has even rank.\n Here, Z_2 is the real Kalton-Peck space.\n (3) The corollary that no hyperplane of the real Kalton-Peck space has a complex structure.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000008", "classification": { "arxiv": [ "math.FA" ], "msc2020": [ "46B03", "46B20", "47A53" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000009:PalomarQseriesPrimeCoefficients.BCoeff_not_multiplicative_witness", "name": "PalomarQseriesPrimeCoefficients.BCoeff_not_multiplicative_witness", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000009", "entry_name": "xiangyazi24/palomar-qseries-prime-coefficients", "repository": "https://github.com/xiangyazi24/palomar-qseries-prime-coefficients", "commit": "0df90a92002da988b4cbc439f5a1c76b2759a14c", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "For an explicitly defined difference B of two signed cone theta series over Q(sqrt(5)), Lean proves a closed nonmultiplicativity witness and a complete prime-index classification. For every rational prime p congruent to 1 modulo 10, B((p-1)/10) is one of -2, -1, 1, 2; a norm-negative golden integer generator and its strict fundamental-window normalization exist; and the coefficient has magnitude two exactly when the resulting geometric residue invariant is one modulo three. The invariant is also proved independent of the generator within the explicitly defined signed fundamental-unit orbit.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000009", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11F27", "11R11" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000009:PalomarQseriesPrimeCoefficients.prime_BCoeff_complete_classification", "name": "PalomarQseriesPrimeCoefficients.prime_BCoeff_complete_classification", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000009", "entry_name": "xiangyazi24/palomar-qseries-prime-coefficients", "repository": "https://github.com/xiangyazi24/palomar-qseries-prime-coefficients", "commit": "0df90a92002da988b4cbc439f5a1c76b2759a14c", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "For an explicitly defined difference B of two signed cone theta series over Q(sqrt(5)), Lean proves a closed nonmultiplicativity witness and a complete prime-index classification. For every rational prime p congruent to 1 modulo 10, B((p-1)/10) is one of -2, -1, 1, 2; a norm-negative golden integer generator and its strict fundamental-window normalization exist; and the coefficient has magnitude two exactly when the resulting geometric residue invariant is one modulo three. The invariant is also proved independent of the generator within the explicitly defined signed fundamental-unit orbit.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000009", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11F27", "11R11" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000009:PalomarQseriesPrimeCoefficients.geometric_iota_eq_of_sameIdeal", "name": "PalomarQseriesPrimeCoefficients.geometric_iota_eq_of_sameIdeal", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000009", "entry_name": "xiangyazi24/palomar-qseries-prime-coefficients", "repository": "https://github.com/xiangyazi24/palomar-qseries-prime-coefficients", "commit": "0df90a92002da988b4cbc439f5a1c76b2759a14c", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "For an explicitly defined difference B of two signed cone theta series over Q(sqrt(5)), Lean proves a closed nonmultiplicativity witness and a complete prime-index classification. For every rational prime p congruent to 1 modulo 10, B((p-1)/10) is one of -2, -1, 1, 2; a norm-negative golden integer generator and its strict fundamental-window normalization exist; and the coefficient has magnitude two exactly when the resulting geometric residue invariant is one modulo three. The invariant is also proved independent of the generator within the explicitly defined signed fundamental-unit orbit.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000009", "classification": { "arxiv": [ "math.NT" ], "msc2020": [ "11F27", "11R11" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000010:Palomar.fjp_1_1_isSheafy", "name": "Palomar.fjp_1_1_isSheafy", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000010", "entry_name": "CBirkbeck/uniform-sheafy-tate-domains-lean", "repository": "https://github.com/CBirkbeck/uniform-sheafy-tate-domains-lean", "commit": "93be0af164d6feec98f17326dd6dcef884751fc9", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalisation of *Uniform sheafy Tate rings that are not stably uniform* (Birkbeck–Torzewski), answering Question 7 of Kedlaya's *Nonarchimedean Scottish Book* in the negative: a sheafy uniform Huber pair need not be stably uniform. Buzzard–Verberkmoes and Mihara proved that stably uniform Tate Huber rings are sheafy; whether the converse holds was open.\nThe paper gives two complete Tate algebras over a complete discretely valued field `k` that are uniform, nonnoetherian integral domains with `A° = A₀`, strongly sheafy (every Tate algebra `A⟨X₁, …, Xₙ⟩` is sheafy), and yet not stably uniform, because an explicit rational localisation is not uniform. The first (Theorem 1.1) is the finite-jet pinching algebra `𝓐 = B ×_D C`, the pullback of a Milnor square with `B = k⟨W, Q⟩/(Q²)`, `C = k⟨W, W⁻¹⟩⟨Q⟩`, `D = C/(Q²)` — concretely, the series in `k⟨W, W⁻¹⟩⟨Q⟩` whose `Q⁰`- and `Q¹`-coefficients have nonnegative `W`-support; its bad chart `𝓐⟨W/ϖ⟩ ≅ k⟨X, Q⟩/(Q²)` is non-reduced. The second (Theorem 8.1) is the weighted-parity algebra `𝒜_w`, whose bad chart is an integral domain; the paper proves it for every unbounded weight `w ≥ 1`, the formalisation for `w(n) = n`.\nThis submission certifies the seven conclusions of Theorem 1.1 together with two statements ruling out a vacuous reading of its sheafiness claims, stated self-containedly on Mathlib in `Challenge.lean` over an arbitrary complete ultrametric field with DVR valuation ring; Theorem 8.1 is certified by the repository's in-library configuration. The repository also develops the surrounding theory after Wedhorn's *Adic Spaces*: Huber and Tate rings, the adic spectrum, rational localisations, the structure presheaf, and sheafiness (Definition 8.26), including the strongly noetherian theorem (8.28(b)).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000010", "classification": { "arxiv": [ "math.AG", "math.NT" ], "msc2020": [ "14G22", "12J25", "13J10", "16W80", "14G45" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000010:Palomar.fjp_1_1_isUniform", "name": "Palomar.fjp_1_1_isUniform", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000010", "entry_name": "CBirkbeck/uniform-sheafy-tate-domains-lean", "repository": "https://github.com/CBirkbeck/uniform-sheafy-tate-domains-lean", "commit": "93be0af164d6feec98f17326dd6dcef884751fc9", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalisation of *Uniform sheafy Tate rings that are not stably uniform* (Birkbeck–Torzewski), answering Question 7 of Kedlaya's *Nonarchimedean Scottish Book* in the negative: a sheafy uniform Huber pair need not be stably uniform. Buzzard–Verberkmoes and Mihara proved that stably uniform Tate Huber rings are sheafy; whether the converse holds was open.\nThe paper gives two complete Tate algebras over a complete discretely valued field `k` that are uniform, nonnoetherian integral domains with `A° = A₀`, strongly sheafy (every Tate algebra `A⟨X₁, …, Xₙ⟩` is sheafy), and yet not stably uniform, because an explicit rational localisation is not uniform. The first (Theorem 1.1) is the finite-jet pinching algebra `𝓐 = B ×_D C`, the pullback of a Milnor square with `B = k⟨W, Q⟩/(Q²)`, `C = k⟨W, W⁻¹⟩⟨Q⟩`, `D = C/(Q²)` — concretely, the series in `k⟨W, W⁻¹⟩⟨Q⟩` whose `Q⁰`- and `Q¹`-coefficients have nonnegative `W`-support; its bad chart `𝓐⟨W/ϖ⟩ ≅ k⟨X, Q⟩/(Q²)` is non-reduced. The second (Theorem 8.1) is the weighted-parity algebra `𝒜_w`, whose bad chart is an integral domain; the paper proves it for every unbounded weight `w ≥ 1`, the formalisation for `w(n) = n`.\nThis submission certifies the seven conclusions of Theorem 1.1 together with two statements ruling out a vacuous reading of its sheafiness claims, stated self-containedly on Mathlib in `Challenge.lean` over an arbitrary complete ultrametric field with DVR valuation ring; Theorem 8.1 is certified by the repository's in-library configuration. The repository also develops the surrounding theory after Wedhorn's *Adic Spaces*: Huber and Tate rings, the adic spectrum, rational localisations, the structure presheaf, and sheafiness (Definition 8.26), including the strongly noetherian theorem (8.28(b)).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000010", "classification": { "arxiv": [ "math.AG", "math.NT" ], "msc2020": [ "14G22", "12J25", "13J10", "16W80", "14G45" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000010:Palomar.fjp_1_1_isDomain", "name": "Palomar.fjp_1_1_isDomain", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000010", "entry_name": "CBirkbeck/uniform-sheafy-tate-domains-lean", "repository": "https://github.com/CBirkbeck/uniform-sheafy-tate-domains-lean", "commit": "93be0af164d6feec98f17326dd6dcef884751fc9", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalisation of *Uniform sheafy Tate rings that are not stably uniform* (Birkbeck–Torzewski), answering Question 7 of Kedlaya's *Nonarchimedean Scottish Book* in the negative: a sheafy uniform Huber pair need not be stably uniform. Buzzard–Verberkmoes and Mihara proved that stably uniform Tate Huber rings are sheafy; whether the converse holds was open.\nThe paper gives two complete Tate algebras over a complete discretely valued field `k` that are uniform, nonnoetherian integral domains with `A° = A₀`, strongly sheafy (every Tate algebra `A⟨X₁, …, Xₙ⟩` is sheafy), and yet not stably uniform, because an explicit rational localisation is not uniform. The first (Theorem 1.1) is the finite-jet pinching algebra `𝓐 = B ×_D C`, the pullback of a Milnor square with `B = k⟨W, Q⟩/(Q²)`, `C = k⟨W, W⁻¹⟩⟨Q⟩`, `D = C/(Q²)` — concretely, the series in `k⟨W, W⁻¹⟩⟨Q⟩` whose `Q⁰`- and `Q¹`-coefficients have nonnegative `W`-support; its bad chart `𝓐⟨W/ϖ⟩ ≅ k⟨X, Q⟩/(Q²)` is non-reduced. The second (Theorem 8.1) is the weighted-parity algebra `𝒜_w`, whose bad chart is an integral domain; the paper proves it for every unbounded weight `w ≥ 1`, the formalisation for `w(n) = n`.\nThis submission certifies the seven conclusions of Theorem 1.1 together with two statements ruling out a vacuous reading of its sheafiness claims, stated self-containedly on Mathlib in `Challenge.lean` over an arbitrary complete ultrametric field with DVR valuation ring; Theorem 8.1 is certified by the repository's in-library configuration. The repository also develops the surrounding theory after Wedhorn's *Adic Spaces*: Huber and Tate rings, the adic spectrum, rational localisations, the structure presheaf, and sheafiness (Definition 8.26), including the strongly noetherian theorem (8.28(b)).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000010", "classification": { "arxiv": [ "math.AG", "math.NT" ], "msc2020": [ "14G22", "12J25", "13J10", "16W80", "14G45" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000010:Palomar.fjp_1_1_not_isNoetherianRing", "name": "Palomar.fjp_1_1_not_isNoetherianRing", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000010", "entry_name": "CBirkbeck/uniform-sheafy-tate-domains-lean", "repository": "https://github.com/CBirkbeck/uniform-sheafy-tate-domains-lean", "commit": "93be0af164d6feec98f17326dd6dcef884751fc9", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalisation of *Uniform sheafy Tate rings that are not stably uniform* (Birkbeck–Torzewski), answering Question 7 of Kedlaya's *Nonarchimedean Scottish Book* in the negative: a sheafy uniform Huber pair need not be stably uniform. Buzzard–Verberkmoes and Mihara proved that stably uniform Tate Huber rings are sheafy; whether the converse holds was open.\nThe paper gives two complete Tate algebras over a complete discretely valued field `k` that are uniform, nonnoetherian integral domains with `A° = A₀`, strongly sheafy (every Tate algebra `A⟨X₁, …, Xₙ⟩` is sheafy), and yet not stably uniform, because an explicit rational localisation is not uniform. The first (Theorem 1.1) is the finite-jet pinching algebra `𝓐 = B ×_D C`, the pullback of a Milnor square with `B = k⟨W, Q⟩/(Q²)`, `C = k⟨W, W⁻¹⟩⟨Q⟩`, `D = C/(Q²)` — concretely, the series in `k⟨W, W⁻¹⟩⟨Q⟩` whose `Q⁰`- and `Q¹`-coefficients have nonnegative `W`-support; its bad chart `𝓐⟨W/ϖ⟩ ≅ k⟨X, Q⟩/(Q²)` is non-reduced. The second (Theorem 8.1) is the weighted-parity algebra `𝒜_w`, whose bad chart is an integral domain; the paper proves it for every unbounded weight `w ≥ 1`, the formalisation for `w(n) = n`.\nThis submission certifies the seven conclusions of Theorem 1.1 together with two statements ruling out a vacuous reading of its sheafiness claims, stated self-containedly on Mathlib in `Challenge.lean` over an arbitrary complete ultrametric field with DVR valuation ring; Theorem 8.1 is certified by the repository's in-library configuration. The repository also develops the surrounding theory after Wedhorn's *Adic Spaces*: Huber and Tate rings, the adic spectrum, rational localisations, the structure presheaf, and sheafiness (Definition 8.26), including the strongly noetherian theorem (8.28(b)).", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-22-000010", "classification": { "arxiv": [ "math.AG", "math.NT" ], "msc2020": [ "14G22", "12J25", "13J10", "16W80", "14G45" ] } }, { "id": "palomar:PALOMAR-2026-08-22-000010:Palomar.fjp_1_1_powerBounded_eq_unitBall", "name": "Palomar.fjp_1_1_powerBounded_eq_unitBall", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-22-000010", "entry_name": "CBirkbeck/uniform-sheafy-tate-domains-lean", "repository": "https://github.com/CBirkbeck/uniform-sheafy-tate-domains-lean", "commit": "93be0af164d6feec98f17326dd6dcef884751fc9", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalisation of *Uniform sheafy Tate rings that are not stably uniform* (Birkbeck–Torzewski), answering Question 7 of Kedlaya's *Nonarchimedean Scottish Book* in the negative: a sheafy uniform Huber pair need not be stably uniform. Buzzard–Verberkmoes and Mihara proved that stably uniform Tate Huber rings are sheafy; whether the converse holds was open.\nThe paper gives two complete Tate algebras over a complete discretely valued field `k` that are uniform, nonnoetherian integral domains with `A° = A₀`, strongly sheafy (every Tate algebra `A⟨X₁, …, Xₙ⟩` is sheafy), and yet not stably uniform, because an explicit rational localisation is not uniform. The first (Theorem 1.1) is the finite-jet pinching algebra `𝓐 = B ×_D C`, the pullback of a Milnor square with `B = k⟨W, Q⟩/(Q²)`, `C = k⟨W, W⁻¹⟩⟨Q⟩`, `D = C/(Q²)` — concretely, the series in `k⟨W, W⁻¹⟩⟨Q⟩` whose `Q⁰`- and `Q¹`-coefficients have nonnegative `W`-support; its bad chart `𝓐⟨W/ϖ⟩ ≅ k⟨X, Q⟩/(Q²)` is non-reduced. The second (Theorem 8.1) is the weighted-parity algebra `𝒜_w`, whose bad chart is an integral domain; the paper proves it for every unbounded weight `w ≥ 1`, the formalisation for `w(n) = n`.\nThis submission certifies the seven conclusions of Theorem 1.1 together with two statements ruling out a vacuous reading of its sheafiness claims, stated self-containedly on Mathlib in `Challenge.lean` over an arbitrary complete ultrametric field with DVR valuation ring; Theorem 8.1 is certified by the repository's in-library configuration. 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Buzzard–Verberkmoes and Mihara proved that stably uniform Tate Huber rings are sheafy; whether the converse holds was open.\nThe paper gives two complete Tate algebras over a complete discretely valued field `k` that are uniform, nonnoetherian integral domains with `A° = A₀`, strongly sheafy (every Tate algebra `A⟨X₁, …, Xₙ⟩` is sheafy), and yet not stably uniform, because an explicit rational localisation is not uniform. The first (Theorem 1.1) is the finite-jet pinching algebra `𝓐 = B ×_D C`, the pullback of a Milnor square with `B = k⟨W, Q⟩/(Q²)`, `C = k⟨W, W⁻¹⟩⟨Q⟩`, `D = C/(Q²)` — concretely, the series in `k⟨W, W⁻¹⟩⟨Q⟩` whose `Q⁰`- and `Q¹`-coefficients have nonnegative `W`-support; its bad chart `𝓐⟨W/ϖ⟩ ≅ k⟨X, Q⟩/(Q²)` is non-reduced. The second (Theorem 8.1) is the weighted-parity algebra `𝒜_w`, whose bad chart is an integral domain; the paper proves it for every unbounded weight `w ≥ 1`, the formalisation for `w(n) = n`.\nThis submission certifies the seven conclusions of Theorem 1.1 together with two statements ruling out a vacuous reading of its sheafiness claims, stated self-containedly on Mathlib in `Challenge.lean` over an arbitrary complete ultrametric field with DVR valuation ring; Theorem 8.1 is certified by the repository's in-library configuration. 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Buzzard–Verberkmoes and Mihara proved that stably uniform Tate Huber rings are sheafy; whether the converse holds was open.\nThe paper gives two complete Tate algebras over a complete discretely valued field `k` that are uniform, nonnoetherian integral domains with `A° = A₀`, strongly sheafy (every Tate algebra `A⟨X₁, …, Xₙ⟩` is sheafy), and yet not stably uniform, because an explicit rational localisation is not uniform. The first (Theorem 1.1) is the finite-jet pinching algebra `𝓐 = B ×_D C`, the pullback of a Milnor square with `B = k⟨W, Q⟩/(Q²)`, `C = k⟨W, W⁻¹⟩⟨Q⟩`, `D = C/(Q²)` — concretely, the series in `k⟨W, W⁻¹⟩⟨Q⟩` whose `Q⁰`- and `Q¹`-coefficients have nonnegative `W`-support; its bad chart `𝓐⟨W/ϖ⟩ ≅ k⟨X, Q⟩/(Q²)` is non-reduced. The second (Theorem 8.1) is the weighted-parity algebra `𝒜_w`, whose bad chart is an integral domain; the paper proves it for every unbounded weight `w ≥ 1`, the formalisation for `w(n) = n`.\nThis submission certifies the seven conclusions of Theorem 1.1 together with two statements ruling out a vacuous reading of its sheafiness claims, stated self-containedly on Mathlib in `Challenge.lean` over an arbitrary complete ultrametric field with DVR valuation ring; Theorem 8.1 is certified by the repository's in-library configuration. 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Buzzard–Verberkmoes and Mihara proved that stably uniform Tate Huber rings are sheafy; whether the converse holds was open.\nThe paper gives two complete Tate algebras over a complete discretely valued field `k` that are uniform, nonnoetherian integral domains with `A° = A₀`, strongly sheafy (every Tate algebra `A⟨X₁, …, Xₙ⟩` is sheafy), and yet not stably uniform, because an explicit rational localisation is not uniform. The first (Theorem 1.1) is the finite-jet pinching algebra `𝓐 = B ×_D C`, the pullback of a Milnor square with `B = k⟨W, Q⟩/(Q²)`, `C = k⟨W, W⁻¹⟩⟨Q⟩`, `D = C/(Q²)` — concretely, the series in `k⟨W, W⁻¹⟩⟨Q⟩` whose `Q⁰`- and `Q¹`-coefficients have nonnegative `W`-support; its bad chart `𝓐⟨W/ϖ⟩ ≅ k⟨X, Q⟩/(Q²)` is non-reduced. The second (Theorem 8.1) is the weighted-parity algebra `𝒜_w`, whose bad chart is an integral domain; the paper proves it for every unbounded weight `w ≥ 1`, the formalisation for `w(n) = n`.\nThis submission certifies the seven conclusions of Theorem 1.1 together with two statements ruling out a vacuous reading of its sheafiness claims, stated self-containedly on Mathlib in `Challenge.lean` over an arbitrary complete ultrametric field with DVR valuation ring; Theorem 8.1 is certified by the repository's in-library configuration. 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Buzzard–Verberkmoes and Mihara proved that stably uniform Tate Huber rings are sheafy; whether the converse holds was open.\nThe paper gives two complete Tate algebras over a complete discretely valued field `k` that are uniform, nonnoetherian integral domains with `A° = A₀`, strongly sheafy (every Tate algebra `A⟨X₁, …, Xₙ⟩` is sheafy), and yet not stably uniform, because an explicit rational localisation is not uniform. The first (Theorem 1.1) is the finite-jet pinching algebra `𝓐 = B ×_D C`, the pullback of a Milnor square with `B = k⟨W, Q⟩/(Q²)`, `C = k⟨W, W⁻¹⟩⟨Q⟩`, `D = C/(Q²)` — concretely, the series in `k⟨W, W⁻¹⟩⟨Q⟩` whose `Q⁰`- and `Q¹`-coefficients have nonnegative `W`-support; its bad chart `𝓐⟨W/ϖ⟩ ≅ k⟨X, Q⟩/(Q²)` is non-reduced. The second (Theorem 8.1) is the weighted-parity algebra `𝒜_w`, whose bad chart is an integral domain; the paper proves it for every unbounded weight `w ≥ 1`, the formalisation for `w(n) = n`.\nThis submission certifies the seven conclusions of Theorem 1.1 together with two statements ruling out a vacuous reading of its sheafiness claims, stated self-containedly on Mathlib in `Challenge.lean` over an arbitrary complete ultrametric field with DVR valuation ring; Theorem 8.1 is certified by the repository's in-library configuration. 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This is the covering half of Wyner (1967): the elementary volume bound gives the lower bound on the covering number and a random-covering argument gives the upper bound. The statement is given in Mathlib's own vocabulary, as an asymptotic for Metric.coveringNumber of Metric.sphere 0 1 in EuclideanSpace R (Fin n) at chordal radius 2 sin (theta/2), and introduces no definition of its own; two compared lemmas establish that an angular cap of half-angle theta about a unit vector is exactly the closed chordal ball of that radius, which is what licenses reading the result as a statement about caps. 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This is the covering half of Wyner (1967): the elementary volume bound gives the lower bound on the covering number and a random-covering argument gives the upper bound. The statement is given in Mathlib's own vocabulary, as an asymptotic for Metric.coveringNumber of Metric.sphere 0 1 in EuclideanSpace R (Fin n) at chordal radius 2 sin (theta/2), and introduces no definition of its own; two compared lemmas establish that an angular cap of half-angle theta about a unit vector is exactly the closed chordal ball of that radius, which is what licenses reading the result as a statement about caps. 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This is the covering half of Wyner (1967): the elementary volume bound gives the lower bound on the covering number and a random-covering argument gives the upper bound. The statement is given in Mathlib's own vocabulary, as an asymptotic for Metric.coveringNumber of Metric.sphere 0 1 in EuclideanSpace R (Fin n) at chordal radius 2 sin (theta/2), and introduces no definition of its own; two compared lemmas establish that an angular cap of half-angle theta about a unit vector is exactly the closed chordal ball of that radius, which is what licenses reading the result as a statement about caps. 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This is the covering half of Wyner (1967): the elementary volume bound gives the lower bound on the covering number and a random-covering argument gives the upper bound. The statement is given in Mathlib's own vocabulary, as an asymptotic for Metric.coveringNumber of Metric.sphere 0 1 in EuclideanSpace R (Fin n) at chordal radius 2 sin (theta/2), and introduces no definition of its own; two compared lemmas establish that an angular cap of half-angle theta about a unit vector is exactly the closed chordal ball of that radius, which is what licenses reading the result as a statement about caps. Two further compared theorems rule out the junk values that the N-infinity-valued covering number and the logarithm would otherwise admit.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000001", "classification": { "arxiv": [ "math.MG", "cs.IT" ], "msc2020": [ "52C17", "41A46", "94B65" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000001:MiscMath.Geometry.dist_eq_two_mul_sin_angle_div_two", "name": "MiscMath.Geometry.dist_eq_two_mul_sin_angle_div_two", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000001", "entry_name": "geoconuk/lean-misc-math", "repository": "https://github.com/geoconuk/lean-misc-math", "commit": "4a1b0c422b6b4e0fd98b04b31a5c41b33fcea123", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "For a fixed angular radius theta in (0, pi/2), the least number M(n, theta) of spherical caps of angular radius theta, centred at points of the unit sphere of R^n, whose union is that whole sphere satisfies (1/n) log M(n, theta) -> -log sin theta as n -> infinity. This is the covering half of Wyner (1967): the elementary volume bound gives the lower bound on the covering number and a random-covering argument gives the upper bound. The statement is given in Mathlib's own vocabulary, as an asymptotic for Metric.coveringNumber of Metric.sphere 0 1 in EuclideanSpace R (Fin n) at chordal radius 2 sin (theta/2), and introduces no definition of its own; two compared lemmas establish that an angular cap of half-angle theta about a unit vector is exactly the closed chordal ball of that radius, which is what licenses reading the result as a statement about caps. Two further compared theorems rule out the junk values that the N-infinity-valued covering number and the logarithm would otherwise admit.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000001", "classification": { "arxiv": [ "math.MG", "cs.IT" ], "msc2020": [ "52C17", "41A46", "94B65" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000002:SZ.sqrtTwo_mem_S", "name": "SZ.sqrtTwo_mem_S", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000002", "entry_name": "rwst/Square-Roots", "repository": "https://github.com/rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval (2,3), the only interval where an answer to Dubickas's Problem 5 could live; they are formalized (SZ.exists_mem_MahlerZ_Ioo_two_three) but are not separately compared, being subsumed by the compared Theorem B. The mechanism is not Diophantine approximation but the arithmetic of Cantor sets, and rests on the triviality that √m² = m is an integer: splitting n into even and odd values turns the one orbit into two base-m orbits coupled by x + √m y = σ, and a covering induction — Utz's 1951 theorem on the distance set of the Cantor discontinuum at m = 3, proved here for every base and every modulus — solves that equation inside the restricted-digit set. The same machinery at a general modulus gives, for p ≥ 2 and non-square m ≥ 3 in either of two explicit ranges, a real ξ ≠ 0 with p ∣ ⌊ξ√mⁿ⌋ for every n ≥ 1, hence integral parts that are composite from some point on; and the greedy's branching shows the witnesses for √3 have cardinality 2^ℵ₀, so all but countably many of them are transcendental. The informal account is the companion paper doi:10.13140/RG.2.2.32215.43682. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the collaboration is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency of the development itself.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000002:SZ.sqrtThree_mem_MahlerZ", "name": "SZ.sqrtThree_mem_MahlerZ", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000002", "entry_name": "rwst/Square-Roots", "repository": "https://github.com/rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval (2,3), the only interval where an answer to Dubickas's Problem 5 could live; they are formalized (SZ.exists_mem_MahlerZ_Ioo_two_three) but are not separately compared, being subsumed by the compared Theorem B. The mechanism is not Diophantine approximation but the arithmetic of Cantor sets, and rests on the triviality that √m² = m is an integer: splitting n into even and odd values turns the one orbit into two base-m orbits coupled by x + √m y = σ, and a covering induction — Utz's 1951 theorem on the distance set of the Cantor discontinuum at m = 3, proved here for every base and every modulus — solves that equation inside the restricted-digit set. The same machinery at a general modulus gives, for p ≥ 2 and non-square m ≥ 3 in either of two explicit ranges, a real ξ ≠ 0 with p ∣ ⌊ξ√mⁿ⌋ for every n ≥ 1, hence integral parts that are composite from some point on; and the greedy's branching shows the witnesses for √3 have cardinality 2^ℵ₀, so all but countably many of them are transcendental. The informal account is the companion paper doi:10.13140/RG.2.2.32215.43682. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the collaboration is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency of the development itself.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000002:SZ.sqrtThree_notMem_S", "name": "SZ.sqrtThree_notMem_S", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000002", "entry_name": "rwst/Square-Roots", "repository": "https://github.com/rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval (2,3), the only interval where an answer to Dubickas's Problem 5 could live; they are formalized (SZ.exists_mem_MahlerZ_Ioo_two_three) but are not separately compared, being subsumed by the compared Theorem B. The mechanism is not Diophantine approximation but the arithmetic of Cantor sets, and rests on the triviality that √m² = m is an integer: splitting n into even and odd values turns the one orbit into two base-m orbits coupled by x + √m y = σ, and a covering induction — Utz's 1951 theorem on the distance set of the Cantor discontinuum at m = 3, proved here for every base and every modulus — solves that equation inside the restricted-digit set. The same machinery at a general modulus gives, for p ≥ 2 and non-square m ≥ 3 in either of two explicit ranges, a real ξ ≠ 0 with p ∣ ⌊ξ√mⁿ⌋ for every n ≥ 1, hence integral parts that are composite from some point on; and the greedy's branching shows the witnesses for √3 have cardinality 2^ℵ₀, so all but countably many of them are transcendental. The informal account is the companion paper doi:10.13140/RG.2.2.32215.43682. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the collaboration is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency of the development itself.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000002:SZ.sqrt_natCast_mem_S_iff", "name": "SZ.sqrt_natCast_mem_S_iff", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000002", "entry_name": "rwst/Square-Roots", "repository": "https://github.com/rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval (2,3), the only interval where an answer to Dubickas's Problem 5 could live; they are formalized (SZ.exists_mem_MahlerZ_Ioo_two_three) but are not separately compared, being subsumed by the compared Theorem B. The mechanism is not Diophantine approximation but the arithmetic of Cantor sets, and rests on the triviality that √m² = m is an integer: splitting n into even and odd values turns the one orbit into two base-m orbits coupled by x + √m y = σ, and a covering induction — Utz's 1951 theorem on the distance set of the Cantor discontinuum at m = 3, proved here for every base and every modulus — solves that equation inside the restricted-digit set. The same machinery at a general modulus gives, for p ≥ 2 and non-square m ≥ 3 in either of two explicit ranges, a real ξ ≠ 0 with p ∣ ⌊ξ√mⁿ⌋ for every n ≥ 1, hence integral parts that are composite from some point on; and the greedy's branching shows the witnesses for √3 have cardinality 2^ℵ₀, so all but countably many of them are transcendental. The informal account is the companion paper doi:10.13140/RG.2.2.32215.43682. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the collaboration is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency of the development itself.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000002:SZ.mem_MahlerZ_of_three_le", "name": "SZ.mem_MahlerZ_of_three_le", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000002", "entry_name": "rwst/Square-Roots", "repository": "https://github.com/rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval (2,3), the only interval where an answer to Dubickas's Problem 5 could live; they are formalized (SZ.exists_mem_MahlerZ_Ioo_two_three) but are not separately compared, being subsumed by the compared Theorem B. The mechanism is not Diophantine approximation but the arithmetic of Cantor sets, and rests on the triviality that √m² = m is an integer: splitting n into even and odd values turns the one orbit into two base-m orbits coupled by x + √m y = σ, and a covering induction — Utz's 1951 theorem on the distance set of the Cantor discontinuum at m = 3, proved here for every base and every modulus — solves that equation inside the restricted-digit set. The same machinery at a general modulus gives, for p ≥ 2 and non-square m ≥ 3 in either of two explicit ranges, a real ξ ≠ 0 with p ∣ ⌊ξ√mⁿ⌋ for every n ≥ 1, hence integral parts that are composite from some point on; and the greedy's branching shows the witnesses for √3 have cardinality 2^ℵ₀, so all but countably many of them are transcendental. The informal account is the companion paper doi:10.13140/RG.2.2.32215.43682. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the collaboration is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency of the development itself.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000002:SZ.sqrt_mem_MahlerZ_of_four_le", "name": "SZ.sqrt_mem_MahlerZ_of_four_le", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000002", "entry_name": "rwst/Square-Roots", "repository": "https://github.com/rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval (2,3), the only interval where an answer to Dubickas's Problem 5 could live; they are formalized (SZ.exists_mem_MahlerZ_Ioo_two_three) but are not separately compared, being subsumed by the compared Theorem B. The mechanism is not Diophantine approximation but the arithmetic of Cantor sets, and rests on the triviality that √m² = m is an integer: splitting n into even and odd values turns the one orbit into two base-m orbits coupled by x + √m y = σ, and a covering induction — Utz's 1951 theorem on the distance set of the Cantor discontinuum at m = 3, proved here for every base and every modulus — solves that equation inside the restricted-digit set. The same machinery at a general modulus gives, for p ≥ 2 and non-square m ≥ 3 in either of two explicit ranges, a real ξ ≠ 0 with p ∣ ⌊ξ√mⁿ⌋ for every n ≥ 1, hence integral parts that are composite from some point on; and the greedy's branching shows the witnesses for √3 have cardinality 2^ℵ₀, so all but countably many of them are transcendental. The informal account is the companion paper doi:10.13140/RG.2.2.32215.43682. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the collaboration is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency of the development itself.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000002:SZ.exists_dvd_floor_sqrt", "name": "SZ.exists_dvd_floor_sqrt", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000002", "entry_name": "rwst/Square-Roots", "repository": "https://github.com/rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval (2,3), the only interval where an answer to Dubickas's Problem 5 could live; they are formalized (SZ.exists_mem_MahlerZ_Ioo_two_three) but are not separately compared, being subsumed by the compared Theorem B. The mechanism is not Diophantine approximation but the arithmetic of Cantor sets, and rests on the triviality that √m² = m is an integer: splitting n into even and odd values turns the one orbit into two base-m orbits coupled by x + √m y = σ, and a covering induction — Utz's 1951 theorem on the distance set of the Cantor discontinuum at m = 3, proved here for every base and every modulus — solves that equation inside the restricted-digit set. The same machinery at a general modulus gives, for p ≥ 2 and non-square m ≥ 3 in either of two explicit ranges, a real ξ ≠ 0 with p ∣ ⌊ξ√mⁿ⌋ for every n ≥ 1, hence integral parts that are composite from some point on; and the greedy's branching shows the witnesses for √3 have cardinality 2^ℵ₀, so all but countably many of them are transcendental. The informal account is the companion paper doi:10.13140/RG.2.2.32215.43682. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the collaboration is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency of the development itself.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000002:SZ.exists_eventually_composite_sqrt", "name": "SZ.exists_eventually_composite_sqrt", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000002", "entry_name": "rwst/Square-Roots", "repository": "https://github.com/rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval (2,3), the only interval where an answer to Dubickas's Problem 5 could live; they are formalized (SZ.exists_mem_MahlerZ_Ioo_two_three) but are not separately compared, being subsumed by the compared Theorem B. The mechanism is not Diophantine approximation but the arithmetic of Cantor sets, and rests on the triviality that √m² = m is an integer: splitting n into even and odd values turns the one orbit into two base-m orbits coupled by x + √m y = σ, and a covering induction — Utz's 1951 theorem on the distance set of the Cantor discontinuum at m = 3, proved here for every base and every modulus — solves that equation inside the restricted-digit set. The same machinery at a general modulus gives, for p ≥ 2 and non-square m ≥ 3 in either of two explicit ranges, a real ξ ≠ 0 with p ∣ ⌊ξ√mⁿ⌋ for every n ≥ 1, hence integral parts that are composite from some point on; and the greedy's branching shows the witnesses for √3 have cardinality 2^ℵ₀, so all but countably many of them are transcendental. The informal account is the companion paper doi:10.13140/RG.2.2.32215.43682. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the collaboration is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency of the development itself.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000002:SZ.continuum_le_mk_witnesses_sqrtThree", "name": "SZ.continuum_le_mk_witnesses_sqrtThree", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000002", "entry_name": "rwst/Square-Roots", "repository": "https://github.com/rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval (2,3), the only interval where an answer to Dubickas's Problem 5 could live; they are formalized (SZ.exists_mem_MahlerZ_Ioo_two_three) but are not separately compared, being subsumed by the compared Theorem B. The mechanism is not Diophantine approximation but the arithmetic of Cantor sets, and rests on the triviality that √m² = m is an integer: splitting n into even and odd values turns the one orbit into two base-m orbits coupled by x + √m y = σ, and a covering induction — Utz's 1951 theorem on the distance set of the Cantor discontinuum at m = 3, proved here for every base and every modulus — solves that equation inside the restricted-digit set. The same machinery at a general modulus gives, for p ≥ 2 and non-square m ≥ 3 in either of two explicit ranges, a real ξ ≠ 0 with p ∣ ⌊ξ√mⁿ⌋ for every n ≥ 1, hence integral parts that are composite from some point on; and the greedy's branching shows the witnesses for √3 have cardinality 2^ℵ₀, so all but countably many of them are transcendental. The informal account is the companion paper doi:10.13140/RG.2.2.32215.43682. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the collaboration is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency of the development itself.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000002:SZ.exists_transcendental_witness_sqrtThree", "name": "SZ.exists_transcendental_witness_sqrtThree", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000002", "entry_name": "rwst/Square-Roots", "repository": "https://github.com/rwst/Square-Roots", "commit": "e3184902f13967b9f20f37244b3d1c30610ca846", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper, which answers a problem by Dubickas. Dubickas (Glasgow Math. J. 48 (2006), 331–336) splits the reals α > 1 in two: α lies in 𝒵 when some real ξ ≠ 0 makes every integral part ⌊ξαⁿ⌋, n ≥ 1, even, and in 𝒮 otherwise. At α = 3/2 this is Mahler's question, and open; his Problem 3 asks whether √3 lies in 𝒮 or in 𝒵. This formalization settles that question and the whole square-root slice: √3 ∈ 𝒵 by an explicit ξ = 1.34160899796…, and for an integer m ≥ 2 one has √m ∈ 𝒮 exactly when m = 2. The cells √5, √6, √7, √8 put four explicit points of 𝒵 in the interval (2,3), the only interval where an answer to Dubickas's Problem 5 could live; they are formalized (SZ.exists_mem_MahlerZ_Ioo_two_three) but are not separately compared, being subsumed by the compared Theorem B. The mechanism is not Diophantine approximation but the arithmetic of Cantor sets, and rests on the triviality that √m² = m is an integer: splitting n into even and odd values turns the one orbit into two base-m orbits coupled by x + √m y = σ, and a covering induction — Utz's 1951 theorem on the distance set of the Cantor discontinuum at m = 3, proved here for every base and every modulus — solves that equation inside the restricted-digit set. The same machinery at a general modulus gives, for p ≥ 2 and non-square m ≥ 3 in either of two explicit ranges, a real ξ ≠ 0 with p ∣ ⌊ξ√mⁿ⌋ for every n ≥ 1, hence integral parts that are composite from some point on; and the greedy's branching shows the witnesses for √3 have cardinality 2^ℵ₀, so all but countably many of them are transcendental. The informal account is the companion paper doi:10.13140/RG.2.2.32215.43682. Credit divides as follows, in contributor terms rather than bibliographic ones: the mathematical discovery is the Fable-5 agent's; the formalization is the Fable-5 and Opus-5 agents'; the agents also drafted the prose of the companion paper, which the author revised; the direction and the review are the author's, who is responsible for the mathematical content. Bibliographic authorship of the paper is the author's alone — the collaboration is credited in its Acknowledgements, not in its byline, as is usual for an AI tool. There is no sorry outside the deliberate placeholders in Challenge.lean, no project-defined axiom, no native_decide, and Mathlib is the only Lake dependency of the development itself.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K16", "11A63", "28A80", "11B83" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000003:SquareMaximalityPalomar.squareMaximality", "name": "SquareMaximalityPalomar.squareMaximality", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000003", "entry_name": "novou/square-maximality-palomar", "repository": "https://github.com/novou/square-maximality-palomar", "commit": "b1d30cdb3334aa7bff7834b22b342393cdc97df8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalization of square maximality for spanning trees of induced subgraphs of the square lattice. For every positive integer n and every finite set S of n^2 points in Z^2, the induced lattice graph on S has at most as many spanning trees as the n by n square, with equality exactly when S is a translate of that square. The development also proves the corresponding strict and equality statements for arbitrary spanning subgraphs, together with the finite Matrix--Tree theorem and the proof chain used by the paper.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000003", "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C05", "05C30", "05C35" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000003:SquareMaximalityPalomar.equality_iff_latticeSquare", "name": "SquareMaximalityPalomar.equality_iff_latticeSquare", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000003", "entry_name": "novou/square-maximality-palomar", "repository": "https://github.com/novou/square-maximality-palomar", "commit": "b1d30cdb3334aa7bff7834b22b342393cdc97df8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 formalization of square maximality for spanning trees of induced subgraphs of the square lattice. For every positive integer n and every finite set S of n^2 points in Z^2, the induced lattice graph on S has at most as many spanning trees as the n by n square, with equality exactly when S is a translate of that square. The development also proves the corresponding strict and equality statements for arbitrary spanning subgraphs, together with the finite Matrix--Tree theorem and the proof chain used by the paper.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000003", "classification": { "arxiv": [ "math.CO" ], "msc2020": [ "05C05", "05C30", "05C35" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000004:SparseFock.MainTheorem.fully_independent_sparseStack_matrix", "name": "SparseFock.MainTheorem.fully_independent_sparseStack_matrix", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000004", "entry_name": "DiarHaidary/Nelson-Nguyen-conjecture-upper-bound-palomar", "repository": "https://github.com/DiarHaidary/Nelson-Nguyen-conjecture-upper-bound-palomar", "commit": "3fdbf2cbb26c10c18bc799fa63fbd3a768f6f0e2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Lean 4 formalization of three sparse oblivious-subspace-embedding results at the Nelson--Nguyen parameter scale: the fully independent signed-hash SparseStack construction, independent uniform signed exact-s columns, and a limited-independence transfer for finite laws whose combined hash-sign selector marginals agree with SparseStack through order 4q. At the stated rounded parameters the selected theorems give failure probability at most delta, with explicit strict bounds s < 1356 q / epsilon and m < 306456 (d + q) / epsilon^2 where those parameter conclusions are packaged. The formalization is machine checked but has not been peer reviewed and makes no priority claim.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000004", "classification": { "arxiv": [ "math.NA", "math.PR" ], "msc2020": [ "65F50", "60B20", "68W20" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000004:SparseFock.MainTheoremAssembly.canonicalB_pos", "name": "SparseFock.MainTheoremAssembly.canonicalB_pos", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000004", "entry_name": "DiarHaidary/Nelson-Nguyen-conjecture-upper-bound-palomar", "repository": "https://github.com/DiarHaidary/Nelson-Nguyen-conjecture-upper-bound-palomar", "commit": "3fdbf2cbb26c10c18bc799fa63fbd3a768f6f0e2", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "Lean 4 formalization of three sparse oblivious-subspace-embedding results at the Nelson--Nguyen parameter scale: the fully independent signed-hash SparseStack construction, independent uniform signed exact-s columns, and a limited-independence transfer for finite laws whose combined hash-sign selector marginals agree with SparseStack through order 4q. At the stated rounded parameters the selected theorems give failure probability at most delta, with explicit strict bounds s < 1356 q / epsilon and m < 306456 (d + q) / epsilon^2 where those parameter conclusions are packaged. The formalization is machine checked but has not been peer reviewed and makes no priority claim.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000004", "classification": { "arxiv": [ "math.NA", "math.PR" ], "msc2020": [ "65F50", "60B20", "68W20" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000005:KoecherAlfsenShultz.orderIso_preservesJordan", "name": "KoecherAlfsenShultz.orderIso_preservesJordan", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000005", "entry_name": "ehrlich-b/euclidean-jordan", "repository": "https://github.com/ehrlich-b/euclidean-jordan", "commit": "58fb886dceab0a3439c9ba3fca37f01073f2a0a4", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Euclidean Jordan algebra is a finite-dimensional real inner-product space carrying a commutative bilinear product with a unit, satisfying the Jordan identity (x*y)*(x*x) = x*(y*(x*x)) and the associativity of the inner product = . This is Faraut-Koranyi's definition (Analysis on Symmetric Cones, III.1). This development formalizes the classical structure theory of these algebras.\nMathlib carries the Jordan identity as the typeclass IsCommJordan together with five operator-commutation lemmas, and nothing beyond that: no idempotent theory, no Peirce decomposition, no spectral theory. This development supplies that layer.\nThe registered results, in four separate Comparator configurations. (1) The Koecher / Alfsen-Shultz theorem: a unital linear order isomorphism of a Euclidean Jordan algebra is a Jordan automorphism, with the cone taken to be the sums-of-squares cone and the product carried as a bundled bilinear map so that the statement needs no ring instance to elaborate. (2) The single-element spectral theorem: every element of a finite-dimensional formally real Jordan algebra is a real linear combination of a complete family of pairwise-orthogonal idempotents. Finite-dimensionality is load-bearing rather than decorative - the polynomial algebra R[X] satisfies every other hypothesis and has no nonconstant spectral resolution. (3) The Jordan trace form tau(x,y) = tr(L_{x*y}): symmetric, associative, and positive definite, constructed from the multiplication alone rather than assumed. Symmetry and associativity are registered at their honest generality, which does not include finite-dimensionality; positive definiteness needs the spectral theorem and therefore does. (4) The frame Peirce decomposition: for a Jordan frame p_1, ..., p_n the algebra is the internal direct sum of the blocks V_ij over unordered pairs, together with the Faraut-Koranyi table entry dim V_ii = 1.\nDirection and scope. Every registered theorem is proved outright from the stated hypotheses; none is conditional on a cited result. The development also proves, but does not register here, Albert's power-associativity theorem (x^(m+1) * x^(n+1) = x^(m+n+2), with no unit, finite dimension, formal reality or positivity assumed), which is registered from a separate development of the author's and is deliberately not re-registered; the existence of a Jordan frame in any nontrivial finite-dimensional Euclidean Jordan algebra; the associativity of the subalgebra generated by a single element; the Peirce subalgebras J_2(c) and J_0(c) as Euclidean Jordan algebras in their own right; and an order-unit-space structure with the sums-of-squares cone, proved to coincide with the Loewner order on the concrete carrier.\nWhat is NOT here. Coordinatization is absent: the Jordan-von Neumann-Wigner route continues by making the off-diagonal block a composition algebra and applying Hurwitz's theorem to land J = H_n(K), and that step depends on the composition-algebra classification, which lives in a separate repository. This development therefore stops at the connectors and the frame-block structure. There is no functional calculus on the spectral resolution. Frame conjugacy (Faraut-Koranyi IV.2.5) is not proved - the development sidesteps it by carrying a Jordan frame as data rather than reasoning about the rank as a number. The classification of simple Euclidean Jordan algebras is not proved, and no statement about the exceptional algebra H_3(O) appears. Formal reality and the associativity of the inner product are equivalent over R in finite dimension, but that equivalence is not formalized, so both appear as hypotheses.\nThe abstract theory has a live carrier: H_n(K) for K = R or C, the Hermitian matrices under A*B = (AB+BA)/2, is an instance of the EuclideanJordanAlgebra class, checked by instance synthesis in the mechanical audit rather than by inspection. The check is finer than nonemptiness of the class: every Peirce rule mentioning the half-eigenspace would be vacuously true on a carrier where that space is trivial, and it is trivial for the two idempotents one reaches for first, so a witness with nonzero half-eigenspace is exhibited on H_2(C).\nEvery declaration in the tracked tree has an axiom closure contained in Lean's three core axioms (propext, Classical.choice, Quot.sound). The tree carries no custom axiom, no sorry and no native_decide.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000005", "classification": { "arxiv": [ "math.RA", "math.OA", "math.FA" ], "msc2020": [ "17C20", "17C27", "17C37", "17C65", "17A15", "46L70" ] } }, { "id": "palomar:PALOMAR-2026-08-23-000006:JordanSpectral.spectral_resolution_bilinear", "name": "JordanSpectral.spectral_resolution_bilinear", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-23-000006", "entry_name": "ehrlich-b/euclidean-jordan", "repository": "https://github.com/ehrlich-b/euclidean-jordan", "commit": "58fb886dceab0a3439c9ba3fca37f01073f2a0a4", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Euclidean Jordan algebra is a finite-dimensional real inner-product space carrying a commutative bilinear product with a unit, satisfying the Jordan identity (x*y)*(x*x) = x*(y*(x*x)) and the associativity of the inner product = . This is Faraut-Koranyi's definition (Analysis on Symmetric Cones, III.1). This development formalizes the classical structure theory of these algebras.\nMathlib carries the Jordan identity as the typeclass IsCommJordan together with five operator-commutation lemmas, and nothing beyond that: no idempotent theory, no Peirce decomposition, no spectral theory. This development supplies that layer.\nThe registered results, in four separate Comparator configurations. (1) The Koecher / Alfsen-Shultz theorem: a unital linear order isomorphism of a Euclidean Jordan algebra is a Jordan automorphism, with the cone taken to be the sums-of-squares cone and the product carried as a bundled bilinear map so that the statement needs no ring instance to elaborate. (2) The single-element spectral theorem: every element of a finite-dimensional formally real Jordan algebra is a real linear combination of a complete family of pairwise-orthogonal idempotents. Finite-dimensionality is load-bearing rather than decorative - the polynomial algebra R[X] satisfies every other hypothesis and has no nonconstant spectral resolution. (3) The Jordan trace form tau(x,y) = tr(L_{x*y}): symmetric, associative, and positive definite, constructed from the multiplication alone rather than assumed. Symmetry and associativity are registered at their honest generality, which does not include finite-dimensionality; positive definiteness needs the spectral theorem and therefore does. (4) The frame Peirce decomposition: for a Jordan frame p_1, ..., p_n the algebra is the internal direct sum of the blocks V_ij over unordered pairs, together with the Faraut-Koranyi table entry dim V_ii = 1.\nDirection and scope. Every registered theorem is proved outright from the stated hypotheses; none is conditional on a cited result. The development also proves, but does not register here, Albert's power-associativity theorem (x^(m+1) * x^(n+1) = x^(m+n+2), with no unit, finite dimension, formal reality or positivity assumed), which is registered from a separate development of the author's and is deliberately not re-registered; the existence of a Jordan frame in any nontrivial finite-dimensional Euclidean Jordan algebra; the associativity of the subalgebra generated by a single element; the Peirce subalgebras J_2(c) and J_0(c) as Euclidean Jordan algebras in their own right; and an order-unit-space structure with the sums-of-squares cone, proved to coincide with the Loewner order on the concrete carrier.\nWhat is NOT here. Coordinatization is absent: the Jordan-von Neumann-Wigner route continues by making the off-diagonal block a composition algebra and applying Hurwitz's theorem to land J = H_n(K), and that step depends on the composition-algebra classification, which lives in a separate repository. This development therefore stops at the connectors and the frame-block structure. There is no functional calculus on the spectral resolution. Frame conjugacy (Faraut-Koranyi IV.2.5) is not proved - the development sidesteps it by carrying a Jordan frame as data rather than reasoning about the rank as a number. The classification of simple Euclidean Jordan algebras is not proved, and no statement about the exceptional algebra H_3(O) appears. Formal reality and the associativity of the inner product are equivalent over R in finite dimension, but that equivalence is not formalized, so both appear as hypotheses.\nThe abstract theory has a live carrier: H_n(K) for K = R or C, the Hermitian matrices under A*B = (AB+BA)/2, is an instance of the EuclideanJordanAlgebra class, checked by instance synthesis in the mechanical audit rather than by inspection. The check is finer than nonemptiness of the class: every Peirce rule mentioning the half-eigenspace would be vacuously true on a carrier where that space is trivial, and it is trivial for the two idempotents one reaches for first, so a witness with nonzero half-eigenspace is exhibited on H_2(C).\nEvery declaration in the tracked tree has an axiom closure contained in Lean's three core axioms (propext, Classical.choice, Quot.sound). The tree carries no custom axiom, no sorry and no native_decide.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-23-000006", "classification": { "arxiv": [ "math.RA", "math.OA", "math.FA" ], "msc2020": [ "17C20", "17C27", "17C37", "17C65", "17A15", "46L70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.theorem_8_8", "name": "Scott1980.Neighborhood.theorem_8_8", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.NeighborhoodSystem.Element.ext", "name": "Scott1980.Neighborhood.NeighborhoodSystem.Element.ext", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.NeighborhoodSystem.element_le_refl", "name": "Scott1980.Neighborhood.NeighborhoodSystem.element_le_refl", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.NeighborhoodSystem.element_le_trans", "name": "Scott1980.Neighborhood.NeighborhoodSystem.element_le_trans", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.NeighborhoodSystem.element_le_antisymm", "name": "Scott1980.Neighborhood.NeighborhoodSystem.element_le_antisymm", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.U_master_mem", "name": "Scott1980.Neighborhood.U_master_mem", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.UMaster_nonempty", "name": "Scott1980.Neighborhood.UMaster_nonempty", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.U_inter_mem", "name": "Scott1980.Neighborhood.U_inter_mem", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.U_sub_master", "name": "Scott1980.Neighborhood.U_sub_master", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.Example15.P4_master_mem", "name": "Scott1980.Neighborhood.Example15.P4_master_mem", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.Example15.P4_master_nonempty", "name": "Scott1980.Neighborhood.Example15.P4_master_nonempty", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.Example15.P4_inter_mem", "name": "Scott1980.Neighborhood.Example15.P4_inter_mem", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.Example15.P4_sub_master", "name": "Scott1980.Neighborhood.Example15.P4_sub_master", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.Example15.P4_embeds", "name": "Scott1980.Neighborhood.Example15.P4_embeds", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.Exercise722.Ssys_master_mem", "name": "Scott1980.Neighborhood.Exercise722.Ssys_master_mem", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.Exercise722.Ssys_master_nonempty", "name": "Scott1980.Neighborhood.Exercise722.Ssys_master_nonempty", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.Exercise722.Ssys_sub_master", "name": "Scott1980.Neighborhood.Exercise722.Ssys_sub_master", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.Exercise722.Ssys_inter_mem", "name": "Scott1980.Neighborhood.Exercise722.Ssys_inter_mem", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000001:Scott1980.Neighborhood.Exercise722.Ssys_embeds", "name": "Scott1980.Neighborhood.Exercise722.Ssys_embeds", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000001", "entry_name": "catskillsresearch/scott1980", "repository": "https://github.com/catskillsresearch/scott1980", "commit": "ef6c05671582b7883ecd40b7081b642ce5351963", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1980 Oxford lectures, published May 1981 as PRG-19 Lectures on a Mathematical Theory of Computation. The principal sourced Palomar result is the first sentence of Theorem 8.8: the neighbourhood system U over [0,1) ⊆ ℚ is universal, in the sense that every countable neighbourhood system D embeds as a subdomain (D ⊴ U). Comparator also selects two new formal corollaries: P4_embeds for Scott's Example 1.5 and Ssys_embeds for Exercise 7.22. Scott defines those systems but does not separately state the embeddings. Challenge.lean fixes the neighbourhood-system vocabulary, including Scott's standing assumption that the master token set Δ is non-empty, Scott's ⊴, and the concrete Definition 8.7 family U. This foundational universality result supplies a concrete host for all countable domains; it is relevant to domain theory, denotational semantics, mathematical logic, and formalized mathematics. The broader Lectures I–VIII development lives in Scott1980/Neighborhood/* and is not the advertised Comparator result. The informal account is arxiv.md; the working transcription is sources/PRG19.md. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000001", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.FLP_cor_one_four_a", "name": "Z32.FLP_cor_one_four_a", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.dubickas_theorem_1", "name": "Z32.dubickas_theorem_1", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_eq_empty_of_lt_sq", "name": "Z32.ZSet_eq_empty_of_lt_sq", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_three_two_third_empty", "name": "Z32.ZSet_three_two_third_empty", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.residue_not_eventually_constant", "name": "Z32.residue_not_eventually_constant", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.exists_residue_change", "name": "Z32.exists_residue_change", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.residue_not_eventually_constant_one", "name": "Z32.residue_not_eventually_constant_one", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_three_two_sixth_3_8", "name": "Z32.ZSet_three_two_sixth_3_8", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_three_two_frontier", "name": "Z32.ZSet_three_two_frontier", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.not_eventually_mem_sixth_3_8", "name": "Z32.not_eventually_mem_sixth_3_8", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_four_three_beyond_line", "name": "Z32.ZSet_four_three_beyond_line", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.dubickas_2008_cor_1_2", "name": "Z32.dubickas_2008_cor_1_2", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.union_seven_twelfths_empty", "name": "Z32.union_seven_twelfths_empty", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.union_two_thirds_empty", "name": "Z32.union_two_thirds_empty", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.union_record_empty", "name": "Z32.union_record_empty", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.union_record_7083_empty", "name": "Z32.union_record_7083_empty", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.two_cell_fifth_empty", "name": "Z32.two_cell_fifth_empty", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.not_forall_abs_sub_round_lt_fifth", "name": "Z32.not_forall_abs_sub_round_lt_fifth", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_five_two_fifth", "name": "Z32.ZSet_five_two_fifth", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_five_two_grid", "name": "Z32.ZSet_five_two_grid", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_seven_two_grid", "name": "Z32.ZSet_seven_two_grid", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_nine_two_grid", "name": "Z32.ZSet_nine_two_grid", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_ten_three_grid", "name": "Z32.ZSet_ten_three_grid", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000002:Z32.ZSet_eleven_three_grid", "name": "Z32.ZSet_eleven_three_grid", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000002", "entry_name": "rwst/Confinement-Certificates", "repository": "https://github.com/rwst/Confinement-Certificates", "commit": "eb810d8f86e49b2b8e058d02a9edac9e5fedafc8", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / Mathlib development of a paper on the emptiness theory of the sets Z_{p/q}(s, s+t) = {ξ > 0 : {ξ(p/q)ⁿ} ∈ [s, s+t) for all n ≥ 0}, the generalized Z-numbers of Flatto, Lagarias and Pollington, at and beyond the window length 1/p. The paper introduces block certificates --- finite integer witnesses, rechecked from scratch by the Lean proof kernel, whose soundness rests on the aperiodicity lemma alone --- and uses them to pass the length-$1/p$ line: $\\Zset_{3/2}(1/6,13/24)=\\emptyset$ (window length $3/8>1/3$), $\\Zset_{3/2}(961/3600,2427/3600)=\\emptyset$ (length $733/1800$), and explicit unions of total length up to $17/24$, against $20/39$ in print. The closed-interval union statement of Dubickas is replicated exactly as printed; no $\\xi\\ne0$ keeps $\\|\\xi(3/2)^{n}\\|<1/5$ for every $n$, with $\\|\\cdot\\|$ the distance to the nearest integer; and thirty windows of length $1/p$ are certified at five bases in the regime $p>q^{2}$ that Dubickas leaves open. There is no sorry anywhere outside the deliberate placeholders in Challenge.lean, no project-defined axiom, and no native_decide: every block certificate is discharged by `c.ok = true := by decide`, i.e. by kernel reduction of integer arithmetic on the certificate's raw data.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000002", "classification": { "arxiv": [ "math.NT", "math.DS" ], "msc2020": [ "11J71", "11K06", "37B10" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.theorem_4_4", "name": "Scott1972.ContinuousLattice.theorem_4_4", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.ScottMap.le_refl'", "name": "Scott1972.ContinuousLattice.ScottMap.le_refl'", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.ScottMap.le_trans'", "name": "Scott1972.ContinuousLattice.ScottMap.le_trans'", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.ScottMap.le_antisymm'", "name": "Scott1972.ContinuousLattice.ScottMap.le_antisymm'", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.ScottMap.lt_iff_le_not_ge'", "name": "Scott1972.ContinuousLattice.ScottMap.lt_iff_le_not_ge'", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.ScottMap.sSupMaps_continuous", "name": "Scott1972.ContinuousLattice.ScottMap.sSupMaps_continuous", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.ScottMap.isLUB_sSup", "name": "Scott1972.ContinuousLattice.ScottMap.isLUB_sSup", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.functionSpaceInclFun_continuous", "name": "Scott1972.ContinuousLattice.functionSpaceInclFun_continuous", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.functionSpaceInclFun_outer_continuous", "name": "Scott1972.ContinuousLattice.functionSpaceInclFun_outer_continuous", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.functionSpaceRetrFun_continuous", "name": "Scott1972.ContinuousLattice.functionSpaceRetrFun_continuous", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.functionSpaceRetrFun_outer_continuous", "name": "Scott1972.ContinuousLattice.functionSpaceRetrFun_outer_continuous", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.functionSpaceRetrIncl", "name": "Scott1972.ContinuousLattice.functionSpaceRetrIncl", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.functionSpaceInclRetrLe", "name": "Scott1972.ContinuousLattice.functionSpaceInclRetrLe", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.inverseLimitLE_refl", "name": "Scott1972.ContinuousLattice.inverseLimitLE_refl", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.inverseLimitLE_trans", "name": "Scott1972.ContinuousLattice.inverseLimitLE_trans", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.inverseLimitLE_antisymm", "name": "Scott1972.ContinuousLattice.inverseLimitLE_antisymm", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.inverseLimitLT_iff_LE_not_GE", "name": "Scott1972.ContinuousLattice.inverseLimitLT_iff_LE_not_GE", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.compatible_sInf", "name": "Scott1972.ContinuousLattice.compatible_sInf", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000003:Scott1972.ContinuousLattice.isGLB_sInf'", "name": "Scott1972.ContinuousLattice.isGLB_sInf'", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000003", "entry_name": "catskillsresearch/scott1972", "repository": "https://github.com/catskillsresearch/scott1972", "commit": "278ba1745b6fb023fc3e9ffe53b7106ce60222c1", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1972 paper *Continuous Lattices* (LNM 274), through the Milner correction (pp. 135–136). The Palomar compared result is Theorem 4.4 as worded in sources/ScottContinLatt1972.md: the inverse limit D_∞ of the recursively defined sequence ⟨D_n, j_n⟩ of function spaces is not only a continuous lattice, but it is also homeomorphic to its own function space [D_∞ → D_∞]. The compared Homeomorph explicitly uses the product/subspace topology on the inverse limit and the pointwise Pi topology on the function space. The full paper development (injectivity, Scott topology, way-below, function spaces, inverse limits) lives in Scott1972/ContinuousLattice/*. The informal account is arxiv.md and sources/ScottContinLatt1972.md in this repository. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000003", "classification": { "arxiv": [ "math.LO", "math.GN" ], "msc2020": [ "06B35", "54H99", "03B40" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000004:hansen_cograph_mahler", "name": "hansen_cograph_mahler", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000004", "entry_name": "jeremypparker/mahler-hansen-cograph", "repository": "https://github.com/jeremypparker/mahler-hansen-cograph", "commit": "72eee5b643fe4f003eea56b8192adb25ba08aa2e", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4/Mathlib formalisation of a sharp Mahler-volume theorem for Hansen polytopes of finite cographs. It proves that the normalized Mahler volume of the Hansen polytope of a cograph is at least one, and exactly one if and only if it is a threshold graph.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000004", "classification": { "arxiv": [ "math.MG", "math.CO" ], "msc2020": [ "52A40", "52B05", "05C75" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000004:hansen_cograph_mahler_eq_one_iff_threshold", "name": "hansen_cograph_mahler_eq_one_iff_threshold", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000004", "entry_name": "jeremypparker/mahler-hansen-cograph", "repository": "https://github.com/jeremypparker/mahler-hansen-cograph", "commit": "72eee5b643fe4f003eea56b8192adb25ba08aa2e", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4/Mathlib formalisation of a sharp Mahler-volume theorem for Hansen polytopes of finite cographs. It proves that the normalized Mahler volume of the Hansen polytope of a cograph is at least one, and exactly one if and only if it is a threshold graph.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000004", "classification": { "arxiv": [ "math.MG", "math.CO" ], "msc2020": [ "52A40", "52B05", "05C75" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.theorem_7_2", "name": "Scott1982.InfoSys.theorem_7_2", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.element_toApproxMap_approxMap_toElement", "name": "Scott1982.InfoSys.element_toApproxMap_approxMap_toElement", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.approxMap_toElement_element_toApproxMap", "name": "Scott1982.InfoSys.approxMap_toElement_element_toApproxMap", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.approxMap_toElement_consistent", "name": "Scott1982.InfoSys.approxMap_toElement_consistent", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.approxMap_toElement_closed", "name": "Scott1982.InfoSys.approxMap_toElement_closed", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.element_toApproxMap_rel_dom", "name": "Scott1982.InfoSys.element_toApproxMap_rel_dom", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.element_toApproxMap_rel_cod", "name": "Scott1982.InfoSys.element_toApproxMap_rel_cod", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.element_toApproxMap_empty_rel", "name": "Scott1982.InfoSys.element_toApproxMap_empty_rel", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.element_toApproxMap_union_right", "name": "Scott1982.InfoSys.element_toApproxMap_union_right", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.element_toApproxMap_mono", "name": "Scott1982.InfoSys.element_toApproxMap_mono", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.functionSystem_con_subset", "name": "Scott1982.InfoSys.functionSystem_con_subset", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.functionSystem_con_sing", "name": "Scott1982.InfoSys.functionSystem_con_sing", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.functionSystem_ent_con", "name": "Scott1982.InfoSys.functionSystem_ent_con", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.functionSystem_ent_bot", "name": "Scott1982.InfoSys.functionSystem_ent_bot", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.functionSystem_ent_refl", "name": "Scott1982.InfoSys.functionSystem_ent_refl", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.functionSystem_ent_trans", "name": "Scott1982.InfoSys.functionSystem_ent_trans", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.Element.ext", "name": "Scott1982.InfoSys.Element.ext", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.element_le_refl", "name": "Scott1982.InfoSys.element_le_refl", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.element_le_trans", "name": "Scott1982.InfoSys.element_le_trans", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.element_le_antisymm", "name": "Scott1982.InfoSys.element_le_antisymm", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.con_empty", "name": "Scott1982.InfoSys.con_empty", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.funion_left_comm", "name": "Scott1982.InfoSys.funion_left_comm", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.funion_left_comm'", "name": "Scott1982.InfoSys.funion_left_comm'", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.funBot_property", "name": "Scott1982.InfoSys.funBot_property", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.Constructive.insert_comm'", "name": "Scott1982.Constructive.insert_comm'", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.Constructive.decidableEq_finset_eq_of_subset", "name": "Scott1982.Constructive.decidableEq_finset_eq_of_subset", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.Constructive.decidableEq_finset_false_of_ne", "name": "Scott1982.Constructive.decidableEq_finset_false_of_ne", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. There is no sorry outside the deliberate holes in Challenge.lean, no project-defined axiom, and Mathlib is the only Lake dependency.", "informal": "", "informal_source": "palomar metadata", "checks": { "upstream": "palomar" }, "upstream_checks": { "comparator": true, "kernels": [ "lean", "nanoda" ], "human_signoff": false, "llm_review": true }, "signoffs": [], "mathlib_conflicts": [], "url": "https://palomar-registry.org/PALOMAR-2026-08-24-000005", "classification": { "arxiv": [ "cs.LO", "math.LO" ], "msc2020": [ "68Q55", "06B35", "03B70" ] } }, { "id": "palomar:PALOMAR-2026-08-24-000005:Scott1982.InfoSys.mkFunToken_property", "name": "Scott1982.InfoSys.mkFunToken_property", "kind": "", "origin": "overlay:palomar", "entry": "PALOMAR-2026-08-24-000005", "entry_name": "catskillsresearch/scott1982", "repository": "https://github.com/catskillsresearch/scott1982", "commit": "d3221eec09505aa75e1b818aabc73f1f04339bdc", "lean": "", "spec_file": "", "impl_module": "", "lines": null, "statement": "", "doc": "A Lean 4 / mathlib formalization of Dana Scott's 1982 ICALP paper *Domains for Denotational Semantics*. The Palomar compared result is only the first sentence of Theorem 7.2: if A and B are information systems, then so is A → B, and the approximable mappings f : A → B are exactly the elements f ∈ |A → B|. Challenge.lean imports Mathlib only and locks Definition 2.1, Definition 2.2, Definition 5.1, and Definition 7.1 (tokens, consistency, and the constructive-witness form of entailment). The remaining apply / curry clauses of the same numbered theorem live in Scott1982/Theorem72.lean and are not Comparator targets. The informal account is arxiv.md; ground truth for theorem wording is sources/Domains_for_Denotational_Semantics.pdf. Lean was written by AI agents under the direction and review of the author. Dana Scott did not participate in or endorse this formalization. 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