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

Proved

OAI.CoulombBarrier.exists_actual_selected_data_constants

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.exists_actual_selected_data_constants of lean/OAI/Analysis/CoulombIonization/RadialBounds/SelectedData.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, open CoulombAtom CoulombAnalysis CoulombObservation):

theorem exists_actual_selected_data_constants :
    ∃ B C : ℝ, 0 < B ∧ 0 ≤ C ∧
      ∀ {ι : Type*} {f : Filter ι} (Z : ι → ℕ) (s : ι → ℝ) (N : ι → ℕ),
        (∀ i, 1 ≤ Z i) → (∀ i, 0 < s i) → Tendsto s f (𝓝 0) →
        Tendsto (fun i => (Z i:ℝ)*(s i)^3) f atTop →
        (∀ i, PriceMinimizes (energy (Z i)) ((s i)^(-4:ℝ)) (N i)) →
        ∀ l : ℝ, 1 ≤ l → ∀ᶠ i in f,
          Nonempty (SelectedQuantumData (Z i) (N i) (s i) l (selectedTailThreshold (s i)) B C selectedTailRadius)

Definitions. The statement is written with OAI.CoulombAtom.PriceMinimizes (definitions bundle OAIQ071C001 of this split); OAI.CoulombBarrier.SelectedQuantumData, OAI.CoulombBarrier.selectedTailRadius and OAI.CoulombBarrier.selectedTailThreshold (definitions bundle OAIQ071C010 of this split); OAI.CoulombAtom.energy (the published bundle Def_CoulombIonization).

Used by. The proofs of OAI_CoulombAtom_actual_priced_deficit_limit and OAI.CoulombAtom.generalized_ionization of this split import 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

section
open MeasureTheory Filter Set Metric
open scoped Topology
open CoulombAtom CoulombAnalysis CoulombObservation
attribute [local irreducible] graphRawLaw physicalObservationLaw jointMasterPosterior

theorem exists_actual_selected_data_constants :
    ∃ B C : ℝ, 0 < B ∧ 0 ≤ C ∧
      ∀ {ι : Type*} {f : Filter ι} (Z : ι → ℕ) (s : ι → ℝ) (N : ι → ℕ),
        (∀ i, 1 ≤ Z i) → (∀ i, 0 < s i) → Tendsto s f (𝓝 0) →
        Tendsto (fun i => (Z i:ℝ)*(s i)^3) f atTop →
        (∀ i, PriceMinimizes (energy (Z i)) ((s i)^(-4:ℝ)) (N i)) →
        ∀ l : ℝ, 1 ≤ l → ∀ᶠ i in f,
          Nonempty (SelectedQuantumData (Z i) (N i) (s i) l (selectedTailThreshold (s i)) B C selectedTailRadius) := by
  sorry

end
open MeasureTheory Filter Set Metric
open scoped Topology
open CoulombAtom CoulombAnalysis

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/SelectedData.lean (declaration OAI.CoulombBarrier.exists_actual_selected_data_constants)

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