import EntropyResolventFullCheeger
import OpenAIKLS500

open MeasureTheory
open scoped ENNReal NNReal
noncomputable section
namespace KLS

/-- The achieved Cheeger conversion coefficient is strictly below the requested threshold. -/
theorem entropyCheegerCoefficient_lt_two : (197/100 : ℝ) < 2 := by norm_num

/-- Explicit improved closed-neighborhood Cheeger bound on the full original law class. -/
theorem admissibleMeasure.cheeger_lower_entropy197_500
    {n : ℕ} (hn : 1 ≤ n) {μ : Measure (Space n)} (hμ : admissibleMeasure μ) :
    ENNReal.ofReal (100/(197*Real.sqrt 500)) ≤ cheegerConstant μ := by
  apply hμ.cheeger_lower_bound_entropy_of_uniform_strong_poincare
    (C := (500 : ℝ≥0)) (by norm_num)
  intro ν hν _
  exact hν.poincare_mem_coupledRankYoung hn

/-- Both full original Cheeger measure classes receive the same numerical bound. -/
theorem fullCheegerVerification_entropy197_500 : FullCheegerVerification := by
  apply fullCheegerVerification_entropy_of_uniform_strong_poincare
    (C := (500 : ℝ≥0)) (by norm_num)
  intro n hn ν hν _
  exact hν.poincare_mem_coupledRankYoung hn

/-- The original KLS proposition follows with the improved Cheeger witness. -/
theorem klsConjecture_entropy197_500 : KLSConjecture := by
  obtain ⟨c,hc,_,hfull⟩ := fullCheegerVerification_entropy197_500
  exact ⟨c,hc,hfull⟩

/-- The exact upstream dimension-free OpenAI endpoint remains the certified theorem. -/
theorem exactOpenAIKLSStatement_entropy197_500 : OAI.LeanBlast.KLS.KLSStatement :=
  OAI.LeanBlast.KLS.klsStatement500

/-- Our fixed witness500 for OpenAI's original Poincare statement, together with
our improved numerical Cheeger theorem on every original admissible law. -/
theorem dimensionFree500_and_cheeger197 :
    (∀ n : ℕ, 1 ≤ n → ∀ ρ : OAI.LeanBlast.KLS.Space n → ℝ,
      OAI.LeanBlast.KLS.IsLogConcaveDensity ρ →
      OAI.LeanBlast.KLS.IsIsotropic (OAI.LeanBlast.KLS.densityMeasure ρ) →
      OAI.LeanBlast.KLS.PoincareBound (OAI.LeanBlast.KLS.densityMeasure ρ) 500) ∧
    (∀ n : ℕ, 1 ≤ n → ∀ μ : Measure (Space n), admissibleMeasure μ →
      ENNReal.ofReal (100/(197*Real.sqrt 500)) ≤ cheegerConstant μ) := by
  exact ⟨OAI.LeanBlast.KLS.poincareBound500,
    fun _ hn _ hμ => hμ.cheeger_lower_entropy197_500 hn⟩

end KLS
end