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

Proved

OAI.CoulombBarrier.actual_outward_barrier_given_state_eventually

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.actual_outward_barrier_given_state_eventually of lean/OAI/Analysis/CoulombIonization/RadialBounds/SelectionBudget.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem actual_outward_barrier_given_state_eventually {c : ℝ} (hc : 0 < c)
    (hcL : c < (10*(100000:ℝ))⁻¹) :
    ∃ ε B C M D : ℝ, 0 < ε ∧ ε < 1 ∧ 0 < B ∧ 2 < M ∧ 0 ≤ D ∧
      ∀ {ι : Type*} {l : Filter ι} (Z : ι → ℕ) (s : ι → ℝ) (N K : ι → ℕ),
        (∀ i, 1 ≤ Z i) → (∀ i, 0 < s i) → Tendsto s l (𝓝 0) →
        Tendsto (fun i => (Z i:ℝ)*(s i)^3) l atTop →
        (∀ i, PriceMinimizes (energy (Z i)) ((s i)^(-4:ℝ)) (N i)) →
        ∀ F : ∀ i, fermionGraph (N i),
          (∀ i, OwnProbabilityTailTiltState (Z i) ((s i)^(-4:ℝ)) (ε*(Z i:ℝ)^(-1/3:ℝ)) (K i)
            (fun j => tinyProbabilityFloor (Z i) ((2:ℝ)^j.val*(ε*(Z i:ℝ)^(-1/3:ℝ)))) 1 (F i)) →
          ∀ᶠ i in l, ∀ J : ℕ, J ≤ K i → (2:ℝ)^J*(ε*(Z i:ℝ)^(-1/3:ℝ)) ≤ s i →
            ∃ a p : (Configuration (N i) × (Fin (K i) × (Fin (N i) × Fin 3) → ℝ)) → TFSpace → ℝ,
              IsNuclearBarrier (physicalObservationLaw (graphRawLaw (F i)) (K i))
                (originalQueryDensity (F i) (ε*(Z i:ℝ)^(-1/3:ℝ)) J c (ε*(Z i:ℝ)^(-1/3:ℝ)) (s i))
                (Z i:ℝ) tfDensityCoefficient B C ((2:ℝ)^J*(ε*(Z i:ℝ)^(-1/3:ℝ))) M
                (5184*(s i)^32+D*(s i)^barrierErrorExponent) a p

Definitions. The statement is written with OAI.CoulombAnalysis.TFSpace, OAI.CoulombAtom.SectorGraph, OAI.CoulombAtom.WeakGraphAmbient, OAI.CoulombAtom.fermionGraph, OAI.CoulombAtom.sectorDirections and OAI.CoulombAtom.weakGraph (definitions bundle OAIQ071C000 of this split); OAI.CoulombAnalysis.tfDensityCoefficient and OAI.CoulombAtom.PriceMinimizes (definitions bundle OAIQ071C001 of this split); OAI.CoulombAtom.tinyProbabilityFloor (definitions bundle OAIQ071C002 of this split); OAI.CoulombAtom.graphRawLaw (definitions bundle OAIQ071C003 of this split); OAI.CoulombBarrier.IsNuclearBarrier and OAI.CoulombBarrier.barrierErrorExponent (definitions bundle OAIQ071C005 of this split); OAI.CoulombAtom.physicalObservationLaw (definitions bundle OAIQ071C006 of this split); OAI.CoulombAtom.OwnProbabilityTailTiltState and OAI.CoulombAtom.originalQueryDensity (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.exists_actual_selected_data_constants of this split imports it.

Preamble

import Mathlib
import Definitions.Def_CoulombIonization
import Definitions.Def_OAIQ071C010

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

noncomputable section

namespace CoulombBarrier

open MeasureTheory Filter Set Metric
open scoped Topology BigOperators
open CoulombAtom CoulombAnalysis CoulombObservation
attribute [local irreducible] graphComponent graphFormVector fermionGraph weakGraph fermionGraphValue
attribute [local instance] physicalObservationLaw_probability

theorem actual_outward_barrier_given_state_eventually {c : ℝ} (hc : 0 < c)
    (hcL : c < (10*(100000:ℝ))⁻¹) :
    ∃ ε B C M D : ℝ, 0 < ε ∧ ε < 1 ∧ 0 < B ∧ 2 < M ∧ 0 ≤ D ∧
      ∀ {ι : Type*} {l : Filter ι} (Z : ι → ℕ) (s : ι → ℝ) (N K : ι → ℕ),
        (∀ i, 1 ≤ Z i) → (∀ i, 0 < s i) → Tendsto s l (𝓝 0) →
        Tendsto (fun i => (Z i:ℝ)*(s i)^3) l atTop →
        (∀ i, PriceMinimizes (energy (Z i)) ((s i)^(-4:ℝ)) (N i)) →
        ∀ F : ∀ i, fermionGraph (N i),
          (∀ i, OwnProbabilityTailTiltState (Z i) ((s i)^(-4:ℝ)) (ε*(Z i:ℝ)^(-1/3:ℝ)) (K i)
            (fun j => tinyProbabilityFloor (Z i) ((2:ℝ)^j.val*(ε*(Z i:ℝ)^(-1/3:ℝ)))) 1 (F i)) →
          ∀ᶠ i in l, ∀ J : ℕ, J ≤ K i → (2:ℝ)^J*(ε*(Z i:ℝ)^(-1/3:ℝ)) ≤ s i →
            ∃ a p : (Configuration (N i) × (Fin (K i) × (Fin (N i) × Fin 3) → ℝ)) → TFSpace → ℝ,
              IsNuclearBarrier (physicalObservationLaw (graphRawLaw (F i)) (K i))
                (originalQueryDensity (F i) (ε*(Z i:ℝ)^(-1/3:ℝ)) J c (ε*(Z i:ℝ)^(-1/3:ℝ)) (s i))
                (Z i:ℝ) tfDensityCoefficient B C ((2:ℝ)^J*(ε*(Z i:ℝ)^(-1/3:ℝ))) M
                (5184*(s i)^32+D*(s i)^barrierErrorExponent) a p := by
  sorry

end CoulombBarrier

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/SelectionBudget.lean (declaration OAI.CoulombBarrier.actual_outward_barrier_given_state_eventually)

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