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

Proved

OAI.StandardMapEntropy.actual_fast_bridge_product

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.actual_fast_bridge_product of lean/OAI/Dynamics/StandardMap/StandardMap/FastBridge.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

lemma actual_fast_bridge_product (k : ℝ) (hc : BridgeScalarControl k)
    (N r : ℕ) (hN : 100 ≤ N) (hr : (r:ℝ) ≤ (N:ℝ)/1000)
    {Rw Rh : ℕ} (aw : Fin Rw → ℤ) (nw : Fin Rw → ℕ) (Fw : (Fin Rw → ℝ) → ℝ)
    (ah : Fin Rh → ℤ) (nh : Fin Rh → ℕ) (Fh : (Fin Rh → ℝ) → ℝ)
    (hnw : ∀ j, 0 < nw j) (hnh : ∀ j, 0 < nh j) (L : NNReal)
    (hFw : LipschitzWith L Fw) (hFh : LipschitzWith L Fh)
    (hlenw : ∀ j, nw j ≤ 2*r) (hlenh : ∀ j, nh j ≤ 2*r)
    (hwinw : ∀ j l, 1 ≤ l → l ≤ nw j → (aw j+(l:ℤ)-1).natAbs ≤ r)
    (hwinh : ∀ j l, 1 ≤ l → l ≤ nh j → (ah j+(l:ℤ)-1-(N:ℤ)).natAbs ≤ r)
    (hW0 : ∀ z, 0 ≤ shortfallObservation k aw nw Fw z)
    (hlog : 1 ≤ Real.log (growthBase k)) (hLM : 16*Real.pi*L ≤ growthBase k)
    (h : ℝ) (hδ : growthBase k^(-(1/10:ℝ)*(N:ℝ)) < h/2) :
    (∫⁻ z in {z | torusFastBridge k z N} ∩ {z | h < shortfallObservation k ah nh Fh z},
      ENNReal.ofReal (shortfallObservation k aw nw Fw z) ∂area) ≤
      fastBridgeConstant*((∫⁻ z, ENNReal.ofReal (shortfallObservation k aw nw Fw z) ∂area)+
        ENNReal.ofReal (growthBase k^(-(1/10:ℝ)*(N:ℝ))))*
          area {z | h/2 < shortfallObservation k ah nh Fh z}

Definitions. The statement is written with OAI.StandardMapEntropy.BridgeScalarControl, OAI.StandardMapEntropy.fastBridgeConstant, OAI.StandardMapEntropy.growthBase, OAI.StandardMapEntropy.shortfallObservation and OAI.StandardMapEntropy.torusFastBridge (definitions bundle OAIQ2St000 of this split); OAI.StandardMapEntropy.Circle, OAI.StandardMapEntropy.Torus and OAI.StandardMapEntropy.area (the published bundle Def_StandardMapEntropy).

Used by. The proofs of OAI.StandardMapEntropy.CriticalScaleSequence.dyadicBridge_aeselected and OAI.StandardMapEntropy.CriticalScaleSequence.weight_middle_null of this split import it.

Preamble

import Mathlib
import Definitions.Def_StandardMapEntropy
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 Metric
open scoped ENNReal Topology Classical
namespace StandardMapEntropy
lemma actual_fast_bridge_product (k : ℝ) (hc : BridgeScalarControl k)
    (N r : ℕ) (hN : 100 ≤ N) (hr : (r:ℝ) ≤ (N:ℝ)/1000)
    {Rw Rh : ℕ} (aw : Fin Rw → ℤ) (nw : Fin Rw → ℕ) (Fw : (Fin Rw → ℝ) → ℝ)
    (ah : Fin Rh → ℤ) (nh : Fin Rh → ℕ) (Fh : (Fin Rh → ℝ) → ℝ)
    (hnw : ∀ j, 0 < nw j) (hnh : ∀ j, 0 < nh j) (L : NNReal)
    (hFw : LipschitzWith L Fw) (hFh : LipschitzWith L Fh)
    (hlenw : ∀ j, nw j ≤ 2*r) (hlenh : ∀ j, nh j ≤ 2*r)
    (hwinw : ∀ j l, 1 ≤ l → l ≤ nw j → (aw j+(l:ℤ)-1).natAbs ≤ r)
    (hwinh : ∀ j l, 1 ≤ l → l ≤ nh j → (ah j+(l:ℤ)-1-(N:ℤ)).natAbs ≤ r)
    (hW0 : ∀ z, 0 ≤ shortfallObservation k aw nw Fw z)
    (hlog : 1 ≤ Real.log (growthBase k)) (hLM : 16*Real.pi*L ≤ growthBase k)
    (h : ℝ) (hδ : growthBase k^(-(1/10:ℝ)*(N:ℝ)) < h/2) :
    (∫⁻ z in {z | torusFastBridge k z N} ∩ {z | h < shortfallObservation k ah nh Fh z},
      ENNReal.ofReal (shortfallObservation k aw nw Fw z) ∂area) ≤
      fastBridgeConstant*((∫⁻ z, ENNReal.ofReal (shortfallObservation k aw nw Fw z) ∂area)+
        ENNReal.ofReal (growthBase k^(-(1/10:ℝ)*(N:ℝ))))*
          area {z | h/2 < shortfallObservation k ah nh Fh z} := by
  sorry
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/FastBridge.lean (declaration OAI.StandardMapEntropy.actual_fast_bridge_product)

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