OAI.SKValue.SmoothTerminal.liftProfile_bound: intermediate lemma from OpenAI, QAOA attains the SK ground-state energy in the thermodynamic-first limit

Proved

OAI.SKValue.SmoothTerminal.liftProfile_bound

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.SKValue.value_consequences, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, QAOA attains the SK ground-state energy in the thermodynamic-first limit, OpenAI Math Release, September 25, 2026, result family 281, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.SKValue.SmoothTerminal.liftProfile_bound of lean/OAI/Probability/SKValue/SKValue/Equations/StrictProfile.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.SKValue, with open MeasureTheory ProbabilityTheory Filter Set, open scoped Topology NNReal ENNReal BigOperators):

lemma SmoothTerminal.liftProfile_bound (W : BrownianSpace) {ψ : ℝ → ℝ} (hψ : SmoothTerminal ψ)
    {l : HeatProfile} (hl : l.Pairwise (fun p r ↦ p.2 ≤ r.2)) (hT : profileTime l=1)
    {ε : ℝ≥0} (hε : 0<ε) (k : ℕ) (x : ℝ) :
    |profileValue ψ (liftProfile ε k l) 0 x-profileValue ψ l 0 x|≤(3/2 : ℝ)*(ε : ℝ)*(k+l.length)

Definitions. The statement is written with OAI.SKValue.HeatProfile, OAI.SKValue.SmoothTerminal, OAI.SKValue.liftProfile, OAI.SKValue.profileTime and OAI.SKValue.profileValue (definitions bundle OAIQ4Sk000 of this split); OAI.SKValue.BrownianSpace (the published bundle Def_SKValue).

Used by. The proof of OAI.SKValueG.weak_profile_guerra_upper of this split imports it.

Preamble

import Mathlib
import Definitions.Def_SKValue
import Definitions.Def_OAIQ4Sk000

Formal statement

section

attribute [-instance] _root_.OAI.SKValueG.expOneNoAtoms _root_.OAI.SKValueG.expOneProbability
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

section

open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators
namespace SKValue

lemma SmoothTerminal.liftProfile_bound (W : BrownianSpace) {ψ : ℝ → ℝ} (hψ : SmoothTerminal ψ)
    {l : HeatProfile} (hl : l.Pairwise (fun p r ↦ p.2 ≤ r.2)) (hT : profileTime l=1)
    {ε : ℝ≥0} (hε : 0<ε) (k : ℕ) (x : ℝ) :
    |profileValue ψ (liftProfile ε k l) 0 x-profileValue ψ l 0 x|≤(3/2 : ℝ)*(ε : ℝ)*(k+l.length) := by
  sorry

end SKValue

end

end OAI
end

Source

OpenAI, QAOA attains the SK ground-state energy in the thermodynamic-first limit, OpenAI Math Release, September 25, 2026, https://github.com/openai/math (result family 281; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026/QAOA-attains-the-SK-ground-state-energy-in-the-thermodynamic-first-limit-September-25-2026.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/SKValue/SKValue/Equations/StrictProfile.lean (declaration OAI.SKValue.SmoothTerminal.liftProfile_bound)

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