OAI.CoulombBarrier.tail_initial_good_event: intermediate lemma from OpenAI, Generalized ionization energies for full Coulomb atoms (result family 263)

Proved

OAI.CoulombBarrier.tail_initial_good_event

by dbenbenn · Oct 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.CoulombAtom.generalized_ionization, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Generalized ionization energies for full Coulomb atoms (result family 263), OpenAI Math Release, September 24, 2026, result family 263, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.CoulombBarrier.tail_initial_good_event of lean/OAI/Analysis/CoulombIonization/RadialBounds/TailInitialPointwiseBarrier.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.CoulombBarrier, with open MeasureTheory Filter, open scoped Topology BigOperators ContDiff, open MeasureTheory Filter Set Metric, open scoped BigOperators ContDiff, open CoulombAtom CoulombAnalysis CoulombObservation):

theorem tail_initial_good_event {Z : ℕ} (hZ : 1 ≤ Z)
    {lam r₀ s c₁ : ℝ} (hlam : 0 < lam) (hr : 0 < r₀) (hr1 : r₀ ≤ 1)
    (hs : 0 < s) (hs1 : s ≤ 1) (hrs : r₀ ≤ s) (hc : 0 < c₁) (hc1 : c₁ ≤ 1/2)
    {N K : ℕ} (hN : PriceMinimizes (energy Z) lam N) {p₀ : Fin (K+1) → ℝ} {δ G : ℝ}
    {F : fermionGraph N} (hF : TailTiltState Z lam r₀ K p₀ δ F)
    (j : Fin (K+1)) (h₀ : 0 < p₀ j) (hp40 : p₀ j ≤ r₀^40)
    (hbudget : 3*(Z:ℝ)*lam+dyadicUniformEventBudget ((2:ℝ)^j.val*r₀) (p₀ j) δ ≤ (Z:ℝ)^(7/3:ℝ))
    (hsmall : (Pauli.globalDensityConstant*((Z:ℝ)^(7/3:ℝ)+(Z:ℝ)^(7/3:ℝ)))^(3/5:ℝ)*
      (8*Real.pi)^(2/5:ℝ)*(4*r₀)^(1/5:ℝ) < (Z:ℝ)/(20*r₀))
    (hinterp : (48*Real.pi*G*c₁⁻¹^3)*r₀^(1-3*masterExponent) ≤ 1/20)
    {g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
    (hG : ∀ z, (g z)^2 ≤ G) (hgn : ∫ z, (g z)^2 = 1)
    (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1) :
    ∃ A : Set (Configuration N × (Fin K × (Fin N × Fin 3) → ℝ)),
      MeasurableSet[observationInformation (fun k : Fin K => dyadicObservationWidth r₀ k) j] A ∧
      (∀ z ∈ A, ∀ y : Space, ‖y‖ ≤ (11/10)*r₀ →
        innerPotential r₀ (jointMasterPosterior (graphRawLaw F)
          (fun k : Fin K => dyadicObservationWidth r₀ k) j c₁ r₀ s g
            (originalDatum (fun k : Fin K => dyadicObservationWidth r₀ k) j z)) y ≤ (Z:ℝ)/(10*r₀)) ∧
      (physicalObservationLaw (graphRawLaw F) K).real Aᶜ ≤ 1728*r₀^37

Definitions. The statement is written with OAI.CoulombAnalysis.IsRadial, OAI.CoulombAtom.SectorGraph, OAI.CoulombAtom.WeakGraphAmbient, OAI.CoulombAtom.fermionGraph, OAI.CoulombAtom.sectorDirections and OAI.CoulombAtom.weakGraph (definitions bundle OAIQ071C000 of this split); OAI.CoulombAtom.PriceMinimizes and OAI.CoulombAtom.masterExponent (definitions bundle OAIQ071C001 of this split); OAI.CoulombObservation.dyadicObservationWidth and OAI.CoulombAtom.Pauli.globalDensityConstant (definitions bundle OAIQ071C002 of this split); OAI.CoulombAtom.graphRawLaw (definitions bundle OAIQ071C003 of this split); OAI.CoulombAtom.dyadicUniformEventBudget (definitions bundle OAIQ071C004 of this split); OAI.CoulombAtom.jointMasterPosterior, OAI.CoulombAtom.observationInformation, OAI.CoulombAtom.originalDatum and OAI.CoulombAtom.physicalObservationLaw (definitions bundle OAIQ071C006 of this split); OAI.CoulombAtom.graphRawLaw_finite and OAI.CoulombAtom.innerPotential (definitions bundle OAIQ071C009 of this split); OAI.CoulombAtom.TailTiltState (definitions bundle OAIQ071C010 of this split); OAI.CoulombAtom.Configuration, OAI.CoulombAtom.Space, OAI.CoulombAtom.Spins and OAI.CoulombAtom.energy (the published bundle Def_CoulombIonization).

Used by. The proof of OAI.CoulombBarrier.actual_outward_barrier_given_state_eventually of this split imports it.

Preamble

import Mathlib
import Definitions.Def_CoulombIonization
import Definitions.Def_OAIQ071C010

Formal statement

section

attribute [-instance] _root_.OAI.CoulombAtom.rotationMeasureHaar
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section

open MeasureTheory Filter
open scoped Topology BigOperators ContDiff

section Work_TailInitialGood_barrier_scope

open MeasureTheory Filter Set Metric
open scoped BigOperators ContDiff

namespace CoulombBarrier
open CoulombAtom CoulombAnalysis CoulombObservation
attribute [local instance] physicalObservationLaw_probability

theorem tail_initial_good_event {Z : ℕ} (hZ : 1 ≤ Z)
    {lam r₀ s c₁ : ℝ} (hlam : 0 < lam) (hr : 0 < r₀) (hr1 : r₀ ≤ 1)
    (hs : 0 < s) (hs1 : s ≤ 1) (hrs : r₀ ≤ s) (hc : 0 < c₁) (hc1 : c₁ ≤ 1/2)
    {N K : ℕ} (hN : PriceMinimizes (energy Z) lam N) {p₀ : Fin (K+1) → ℝ} {δ G : ℝ}
    {F : fermionGraph N} (hF : TailTiltState Z lam r₀ K p₀ δ F)
    (j : Fin (K+1)) (h₀ : 0 < p₀ j) (hp40 : p₀ j ≤ r₀^40)
    (hbudget : 3*(Z:ℝ)*lam+dyadicUniformEventBudget ((2:ℝ)^j.val*r₀) (p₀ j) δ ≤ (Z:ℝ)^(7/3:ℝ))
    (hsmall : (Pauli.globalDensityConstant*((Z:ℝ)^(7/3:ℝ)+(Z:ℝ)^(7/3:ℝ)))^(3/5:ℝ)*
      (8*Real.pi)^(2/5:ℝ)*(4*r₀)^(1/5:ℝ) < (Z:ℝ)/(20*r₀))
    (hinterp : (48*Real.pi*G*c₁⁻¹^3)*r₀^(1-3*masterExponent) ≤ 1/20)
    {g : Space → ℝ} (hg : ContDiff ℝ ∞ g) (hcg : HasCompactSupport g)
    (hG : ∀ z, (g z)^2 ≤ G) (hgn : ∫ z, (g z)^2 = 1)
    (hrad : IsRadial g) (hgs : tsupport g ⊆ ball 0 1) :
    ∃ A : Set (Configuration N × (Fin K × (Fin N × Fin 3) → ℝ)),
      MeasurableSet[observationInformation (fun k : Fin K => dyadicObservationWidth r₀ k) j] A ∧
      (∀ z ∈ A, ∀ y : Space, ‖y‖ ≤ (11/10)*r₀ →
        innerPotential r₀ (jointMasterPosterior (graphRawLaw F)
          (fun k : Fin K => dyadicObservationWidth r₀ k) j c₁ r₀ s g
            (originalDatum (fun k : Fin K => dyadicObservationWidth r₀ k) j z)) y ≤ (Z:ℝ)/(10*r₀)) ∧
      (physicalObservationLaw (graphRawLaw F) K).real Aᶜ ≤ 1728*r₀^37 := by
  sorry

end CoulombBarrier

end Work_TailInitialGood_barrier_scope

open MeasureTheory Filter Set Metric
open scoped Topology ContDiff

end

end OAI
end

Source

OpenAI, Generalized ionization energies for full Coulomb atoms (result family 263), OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 263; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Generalized-ionization-energies-for-full-Coulomb-atoms-September-24-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Analysis/CoulombIonization/RadialBounds/TailInitialPointwiseBarrier.lean (declaration OAI.CoulombBarrier.tail_initial_good_event)

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