OAI.StandardMapEntropy.CriticalScaleSequence.capG_limit_balance: intermediate lemma from OpenAI, Positive Metric Entropy for the Standard Map at Large Parameters

Proved

OAI.StandardMapEntropy.CriticalScaleSequence.capG_limit_balance

by dbenbenn · Oct 26, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.StandardMapEntropy.main_entropy, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Positive Metric Entropy for the Standard Map at Large Parameters, OpenAI Math Release, September 23, 2026, result family 146, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.StandardMapEntropy.CriticalScaleSequence.capG_limit_balance of lean/OAI/Dynamics/StandardMap/StandardMap/WeightBalance.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.StandardMapEntropy.CriticalScaleSequence, with open MeasureTheory Set, open scoped ENNReal BigOperators, open MeasureTheory Set Filter, open scoped ENNReal Topology CompactlySupported Classical):

lemma capG_limit_balance {α:ℝ} (hα:0≤α) :
    Tendsto (fun i => ∫d,capG α d ∂S.multiLaw i) (L.filter:Filter ℕ)
      (𝓝 (∫d:NonAffineArray,capG α d.val ∂L.multi))

Definitions. The statement is written with OAI.StandardMapEntropy.AmbientArray, OAI.StandardMapEntropy.CriticalScaleSequence, OAI.StandardMapEntropy.DistanceArray, OAI.StandardMapEntropy.DyadicTime, OAI.StandardMapEntropy.NonAffineArray, OAI.StandardMapEntropy.affineLocus, OAI.StandardMapEntropy.capG, OAI.StandardMapEntropy.CriticalScaleSequence.LimitLaws and OAI.StandardMapEntropy.CriticalScaleSequence.multiLaw (definitions bundle OAIQ2St000 of this split).

Used by. The proof of OAI.StandardMapEntropy.eventually_meanDeficit_small of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ2St000

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

open MeasureTheory Set
open scoped ENNReal BigOperators

open MeasureTheory Set Filter
open scoped ENNReal Topology CompactlySupported Classical
namespace StandardMapEntropy
namespace CriticalScaleSequence
variable (S:CriticalScaleSequence) (L:S.LimitLaws)
lemma capG_limit_balance {α:ℝ} (hα:0≤α) :
    Tendsto (fun i => ∫d,capG α d ∂S.multiLaw i) (L.filter:Filter ℕ)
      (𝓝 (∫d:NonAffineArray,capG α d.val ∂L.multi)) := by
  sorry
end CriticalScaleSequence
end StandardMapEntropy

end OAI
end

Source

OpenAI, Positive Metric Entropy for the Standard Map at Large Parameters, OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 146; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Positive-Metric-Entropy-for-the-Standard-Map-at-Large-Parameters-September-23-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Dynamics/StandardMap/StandardMap/WeightBalance.lean (declaration OAI.StandardMapEntropy.CriticalScaleSequence.capG_limit_balance)

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