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

Proved

OAI.CriticalStrip.HoneycombBranch.step_turn

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.step_turn of lean/OAI/Combinatorics/StripMass/StripMass/PhysicalPorts.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):

lemma step_turn (N : ℕ) (hN : 1≤N) (s i : ℤ) (x : ExitWord (steppedVertices N s) i) :
    turn x.val=stepTurn i (exitPort (stepBoundaryPorts N hN s) x)

Definitions. The statement is written with OAI.CriticalStrip.HoneycombBranch.ExitWord, OAI.CriticalStrip.HoneycombBranch.IsExit, OAI.CriticalStrip.HoneycombBranch.StepEndpoint, OAI.CriticalStrip.HoneycombBranch.exitPort, OAI.CriticalStrip.HoneycombBranch.stepBoundaryPorts, OAI.CriticalStrip.HoneycombBranch.stepTurn, OAI.CriticalStrip.HoneycombBranch.steppedVertices and OAI.CriticalStrip.HoneycombBranch.turn (definitions bundle OAIQ042S001 of this split).

Used by. The proof of OAI.CriticalStrip.HoneycombBranch.physical_stepped_flux of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ042S001

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

lemma step_turn (N : ℕ) (hN : 1≤N) (s i : ℤ) (x : ExitWord (steppedVertices N s) i) :
    turn x.val=stepTurn i (exitPort (stepBoundaryPorts N hN s) x) := 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/PhysicalPorts.lean (declaration OAI.CriticalStrip.HoneycombBranch.step_turn)

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