OAI.CriticalStrip.homogeneous_arch_coefficient_integral: intermediate lemma from OpenAI, Critical strip-crossing mass on the honeycomb lattice (result family 237)

Proved

OAI.CriticalStrip.homogeneous_arch_coefficient_integral

by dbenbenn · Oct 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.CriticalStrip.critical_strip_mass, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Critical strip-crossing mass on the honeycomb lattice (result family 237), OpenAI Math Release, September 26, 2026, result family 237, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.CriticalStrip.homogeneous_arch_coefficient_integral of lean/OAI/Combinatorics/StripMass/StripMass/ReflectionIntegrals.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.CriticalStrip, with open Polynomial Set MeasureTheory Filter, open scoped Topology BigOperators):

theorem homogeneous_arch_coefficient_integral (N : ℕ) (hN : 1 ≤ N)
    {g : Polynomial ℝ} (hg : MonicProblem N g) :
    (N:ℝ)*C - eta*g.coeff (N-1) =
      (eta/Real.pi)*∫ x in Ioo 0 1, momentIntegrand N x

Definitions. The statement is written with OAI.CriticalStrip.MonicProblem (definitions bundle OAIQ042S001 of this split); OAI.CriticalStrip.momentIntegrand (definitions bundle OAIQ042S004 of this split); OAI.CriticalStrip.C and OAI.CriticalStrip.eta (the published bundle Def_CriticalStripMass).

Used by. The proof of OAI.CriticalStrip.criticalStripMass_of_physical_identities of this split imports it.

Preamble

import Mathlib
import Definitions.Def_CriticalStripMass
import Definitions.Def_OAIQ042S004

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

namespace CriticalStrip

noncomputable section
open Polynomial Set MeasureTheory Filter
open scoped Topology BigOperators

theorem homogeneous_arch_coefficient_integral (N : ℕ) (hN : 1 ≤ N)
    {g : Polynomial ℝ} (hg : MonicProblem N g) :
    (N:ℝ)*C - eta*g.coeff (N-1) =
      (eta/Real.pi)*∫ x in Ioo 0 1, momentIntegrand N x := by
  sorry

end
end CriticalStrip

end OAI
end

Source

OpenAI, Critical strip-crossing mass on the honeycomb lattice (result family 237), OpenAI Math Release, September 26, 2026, https://github.com/openai/math (result family 237; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/StripMass/StripMass/ReflectionIntegrals.lean (declaration OAI.CriticalStrip.homogeneous_arch_coefficient_integral)

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