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

Proved

OAI.SKValue.IsDiffusion.curvatureSlope_hasDerivAt

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.IsDiffusion.curvatureSlope_hasDerivAt of lean/OAI/Probability/SKValue/SKValue/Coercivity/CurvatureMoment.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

lemma IsDiffusion.curvatureSlope_hasDerivAt {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
    (hX:IsDiffusion W γ X) {t c:ℝ} (ht:t∈Ioo (0:ℝ) 1) (hc:ContinuousAt γ.coeff t) (he:γ.coeff t=c):
    HasDerivAt (jetMoment W γ X (JetExpr.curvatureSlope c))
      (∫ z,(iteratedDeriv 3 (deriv (phi W γ t)) (X t z))^2-
        12*γ.coeff t*iteratedDeriv 1 (deriv (phi W γ t)) (X t z)*(iteratedDeriv 2 (deriv (phi W γ t)) (X t z))^2+
        6*(γ.coeff t)^2*(iteratedDeriv 1 (deriv (phi W γ t)) (X t z))^4 ∂W.μ) t

Definitions. The statement is written with OAI.SKValue.jetMoment and OAI.SKValue.JetExpr.curvatureSlope (definitions bundle OAIQ4Sk000 of this split); OAI.SKValue.BrownianSpace, OAI.SKValue.IsDiffusion, OAI.SKValue.OrderParameter and OAI.SKValue.phi (the published bundle Def_SKValue).

Used by. The proof of OAI.SKValue.IsMinimizer.gradient_moment_contact 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 Set Filter
open scoped Topology NNReal
namespace SKValue
lemma IsDiffusion.curvatureSlope_hasDerivAt {W:BrownianSpace} {γ:OrderParameter} {X:ℝ → W.Ω → ℝ}
    (hX:IsDiffusion W γ X) {t c:ℝ} (ht:t∈Ioo (0:ℝ) 1) (hc:ContinuousAt γ.coeff t) (he:γ.coeff t=c):
    HasDerivAt (jetMoment W γ X (JetExpr.curvatureSlope c))
      (∫ z,(iteratedDeriv 3 (deriv (phi W γ t)) (X t z))^2-
        12*γ.coeff t*iteratedDeriv 1 (deriv (phi W γ t)) (X t z)*(iteratedDeriv 2 (deriv (phi W γ t)) (X t z))^2+
        6*(γ.coeff t)^2*(iteratedDeriv 1 (deriv (phi W γ t)) (X t z))^4 ∂W.μ) t := 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/Coercivity/CurvatureMoment.lean (declaration OAI.SKValue.IsDiffusion.curvatureSlope_hasDerivAt)

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