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

Proved

OAI.CriticalStrip.positiveLaw_inverseQuarterSum_growth_upper

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.positiveLaw_inverseQuarterSum_growth_upper of lean/OAI/Combinatorics/StripMass/StripMass/MomentGrowth.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem positiveLaw_inverseQuarterSum_growth_upper (n : ℕ) :
    (∫ y, inverseQuarterStatistic (n+1) y ∂positiveLaw (n+1)) ≤
      (2*(64:ℝ)^(1/4:ℝ)*referenceUpperConstant) * ((n:ℝ)+1)^(3/4:ℝ)

Definitions. The statement is written with OAI.CriticalStrip.inverseQuarterStatistic and OAI.CriticalStrip.referenceUpperConstant (definitions bundle OAIQ042S001 of this split); OAI.CriticalStrip.positiveLaw (definitions bundle OAIQ042S004 of this split).

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

Preamble

import Mathlib
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 scoped BigOperators
open Set MeasureTheory
theorem positiveLaw_inverseQuarterSum_growth_upper (n : ℕ) :
    (∫ y, inverseQuarterStatistic (n+1) y ∂positiveLaw (n+1)) ≤
      (2*(64:ℝ)^(1/4:ℝ)*referenceUpperConstant) * ((n:ℝ)+1)^(3/4:ℝ) := 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/MomentGrowth.lean (declaration OAI.CriticalStrip.positiveLaw_inverseQuarterSum_growth_upper)

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