OAI.CriticalStrip.HoneycombBranch.finite_split: intermediate lemma from OpenAI, Critical strip-crossing mass on the honeycomb lattice (result family 237)

Proved

OAI.CriticalStrip.HoneycombBranch.finite_split

by dbenbenn · Oct 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.CriticalStrip.critical_strip_mass, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Critical strip-crossing mass on the honeycomb lattice (result family 237), OpenAI Math Release, September 26, 2026, result family 237, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.CriticalStrip.HoneycombBranch.finite_split of lean/OAI/Combinatorics/StripMass/StripMass/FluxWindow.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.CriticalStrip.HoneycombBranch, with open scoped Classical BigOperators, open FiniteBranchTree Filter Topology):

lemma finite_split {D : Vertex→Prop} {H : ℕ} {i : ℤ}
    (hD : ∀v,D v→0≤v.2.1 ∧ v.2.1<(H:ℤ)) (m : ℕ) :
    (∑l∈physicalExits D H i m,wordWeight factor l)+
    (∑l∈sideExits D H i m,wordWeight factor l)=1

Definitions. The statement is written with OAI.CriticalStrip.FiniteBranchTree.wordWeight, OAI.CriticalStrip.HoneycombBranch.Vertex, OAI.CriticalStrip.HoneycombBranch.physicalExits and OAI.CriticalStrip.HoneycombBranch.sideExits (definitions bundle OAIQ042S001 of this split); OAI.CriticalStrip.HoneycombBranch.factor (definitions bundle OAIQ042S005 of this split).

Used by. The proofs of OAI.CriticalStrip.HoneycombBranch.physical_stepped_flux and OAI.CriticalStrip.critical_strip_mass of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ042S005

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

namespace CriticalStrip.HoneycombBranch
noncomputable section
open scoped Classical BigOperators
open FiniteBranchTree Filter Topology

lemma finite_split {D : Vertex→Prop} {H : ℕ} {i : ℤ}
    (hD : ∀v,D v→0≤v.2.1 ∧ v.2.1<(H:ℤ)) (m : ℕ) :
    (∑l∈physicalExits D H i m,wordWeight factor l)+
    (∑l∈sideExits D H i m,wordWeight factor l)=1 := by
  sorry

end

end CriticalStrip.HoneycombBranch

end OAI
end

Source

OpenAI, Critical strip-crossing mass on the honeycomb lattice (result family 237), OpenAI Math Release, September 26, 2026, https://github.com/openai/math (result family 237; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Critical-strip-crossing-mass-on-the-honeycomb-lattice-September-26-2026/main.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/StripMass/StripMass/FluxWindow.lean (declaration OAI.CriticalStrip.HoneycombBranch.finite_split)

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