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

Proved

OAI.StandardMapEntropy.actual_compact_graph_labels_full

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_compact_graph_labels_full of lean/OAI/Dynamics/StandardMap/StandardMap/TransportGraphs.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 Set Filter Metric, open scoped Topology):

lemma actual_compact_graph_labels_full (k : ℝ) (n : ℕ) (hn : 1 ≤ n) (hk : 0 ≤ k)
    (hq : growthBase k^(-(3/5:ℝ)) ≤ 1/2)
    (hsmall : (384*Real.pi)*growthBase k^(-(7/10:ℝ)) ≤ 1/2)
    (S : Set (ℝ × ℝ)) (hs : IsCompact S)
    (ht : ∀ z ∈ S, tSolution (orbitCoefficient k z.1 z.2) (n+1) ≠ 0)
    (hU : ∀ z ∈ S, ∀ p, 1 ≤ p → p ≤ n →
      |dirichletSolution (orbitCoefficient k z.1 z.2) (n+1) p| ≤
        12*growthBase k^(-(9/10:ℝ)*(p:ℝ)))
    (l u yc : ℝ) (hyc : yc ∈ Icc l u) (hlen : u-l < 1/4)
    (hSp : ∀ z ∈ S, z.1 ∈ Icc l u) :
    ∃ E : Set ℝ, ∃ X : ℝ × ℝ → ℝ,
      IsCompact E ∧ ContinuousOn X (E ×ˢ Icc l u) ∧
      (∀ s ∈ E, X (s,yc)=s) ∧
      (∀ z ∈ S, ∃ s ∈ E, X (s,z.1)=z.2) ∧
      InjOn (fun w => (X w,w.2)) (E ×ˢ Icc l u) ∧
      (∀ s ∈ E, ∀ y ∈ Icc l u,
        liftedOrbit k y (X (s,y)) (n+1)=liftedOrbit k yc s (n+1) ∧
        tSolution (orbitCoefficient k y (X (s,y))) (n+1) ≠ 0 ∧
        HasStrictDerivAt (fun v => X (s,v)) (boundaryValue (orbitCoefficient k y (X (s,y))) (n+1)) y ∧
        |boundaryValue (orbitCoefficient k y (X (s,y))) (n+1)| ≤ 24/growthBase k^((4/5:ℝ)) ∧
        Real.exp (-1152*Real.pi) ≤
          |tSolution (orbitCoefficient k yc s) (n+1)|/
          |tSolution (orbitCoefficient k y (X (s,y))) (n+1)| ∧
        |tSolution (orbitCoefficient k yc s) (n+1)|/
          |tSolution (orbitCoefficient k y (X (s,y))) (n+1)| ≤ Real.exp (1152*Real.pi)) ∧
      (∀ s ∈ E, ∃ z ∈ S, ∀ y ∈ Icc l u,
        liftedOrbit k y (X (s,y)) (n+1)=liftedOrbit k z.1 z.2 (n+1) ∧
        |X (s,y)-z.2| ≤ (24/growthBase k^((4/5:ℝ)))*|y-z.1| ∧
        ∀ i : Fin n, |liftedOrbit k y (X (s,y)) (i+1)-liftedOrbit k z.1 z.2 (i+1)| ≤
          8/growthBase k^((4/5:ℝ)*((i:ℝ)+1)))

Definitions. The statement is written with OAI.StandardMapEntropy.boundaryValue, OAI.StandardMapEntropy.dirichletSolution, OAI.StandardMapEntropy.growthBase, OAI.StandardMapEntropy.liftedOrbit, OAI.StandardMapEntropy.orbitCoefficient and OAI.StandardMapEntropy.tSolution (definitions bundle OAIQ2St000 of this split).

Used by. The proof of OAI.StandardMapEntropy.actual_fast_bridge_product 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 Set Filter Metric
open scoped Topology
namespace StandardMapEntropy
lemma actual_compact_graph_labels_full (k : ℝ) (n : ℕ) (hn : 1 ≤ n) (hk : 0 ≤ k)
    (hq : growthBase k^(-(3/5:ℝ)) ≤ 1/2)
    (hsmall : (384*Real.pi)*growthBase k^(-(7/10:ℝ)) ≤ 1/2)
    (S : Set (ℝ × ℝ)) (hs : IsCompact S)
    (ht : ∀ z ∈ S, tSolution (orbitCoefficient k z.1 z.2) (n+1) ≠ 0)
    (hU : ∀ z ∈ S, ∀ p, 1 ≤ p → p ≤ n →
      |dirichletSolution (orbitCoefficient k z.1 z.2) (n+1) p| ≤
        12*growthBase k^(-(9/10:ℝ)*(p:ℝ)))
    (l u yc : ℝ) (hyc : yc ∈ Icc l u) (hlen : u-l < 1/4)
    (hSp : ∀ z ∈ S, z.1 ∈ Icc l u) :
    ∃ E : Set ℝ, ∃ X : ℝ × ℝ → ℝ,
      IsCompact E ∧ ContinuousOn X (E ×ˢ Icc l u) ∧
      (∀ s ∈ E, X (s,yc)=s) ∧
      (∀ z ∈ S, ∃ s ∈ E, X (s,z.1)=z.2) ∧
      InjOn (fun w => (X w,w.2)) (E ×ˢ Icc l u) ∧
      (∀ s ∈ E, ∀ y ∈ Icc l u,
        liftedOrbit k y (X (s,y)) (n+1)=liftedOrbit k yc s (n+1) ∧
        tSolution (orbitCoefficient k y (X (s,y))) (n+1) ≠ 0 ∧
        HasStrictDerivAt (fun v => X (s,v)) (boundaryValue (orbitCoefficient k y (X (s,y))) (n+1)) y ∧
        |boundaryValue (orbitCoefficient k y (X (s,y))) (n+1)| ≤ 24/growthBase k^((4/5:ℝ)) ∧
        Real.exp (-1152*Real.pi) ≤
          |tSolution (orbitCoefficient k yc s) (n+1)|/
          |tSolution (orbitCoefficient k y (X (s,y))) (n+1)| ∧
        |tSolution (orbitCoefficient k yc s) (n+1)|/
          |tSolution (orbitCoefficient k y (X (s,y))) (n+1)| ≤ Real.exp (1152*Real.pi)) ∧
      (∀ s ∈ E, ∃ z ∈ S, ∀ y ∈ Icc l u,
        liftedOrbit k y (X (s,y)) (n+1)=liftedOrbit k z.1 z.2 (n+1) ∧
        |X (s,y)-z.2| ≤ (24/growthBase k^((4/5:ℝ)))*|y-z.1| ∧
        ∀ i : Fin n, |liftedOrbit k y (X (s,y)) (i+1)-liftedOrbit k z.1 z.2 (i+1)| ≤
          8/growthBase k^((4/5:ℝ)*((i:ℝ)+1))) := 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/TransportGraphs.lean (declaration OAI.StandardMapEntropy.actual_compact_graph_labels_full)

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