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

Proved

OAI.CriticalStrip.Diagram.physical_augmented_stationary

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.Diagram.physical_augmented_stationary of lean/OAI/Combinatorics/StripMass/StripMass/NorthMass.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.CriticalStrip.Diagram, with open scoped Classical BigOperators, open _root_.Matrix Filter Topology):

lemma physical_augmented_stationary (N : ℕ) :
    (restrictedColumn (SewnRectangle.leftWeights N)).mulVec
        (fun p=>angleAugmented (fun _ : Fin N=>lambda) p.val)+
      injectionColumn (SewnRectangle.leftWeights N)=
      fun p=>angleAugmented (fun _ : Fin N=>lambda) p.val

Definitions. The statement is written with OAI.CriticalStrip.Diagram.NonemptyPairing, OAI.CriticalStrip.Diagram.Pairing, OAI.CriticalStrip.Diagram.empty, OAI.CriticalStrip.Diagram.instDecidableEqPairing, OAI.CriticalStrip.Diagram.instFintypePairing and OAI.CriticalStrip.Diagram.lambda (definitions bundle OAIQ042S000 of this split); OAI.CriticalStrip.Diagram.injectionColumn, OAI.CriticalStrip.Diagram.restrictedColumn and OAI.CriticalStrip.SewnRectangle.leftWeights (definitions bundle OAIQ042S002 of this split); OAI.CriticalStrip.Diagram.angleAugmented (definitions bundle OAIQ042S015 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_OAIQ042S015

Formal statement

section

attribute [-instance] _root_.OAI.CriticalStrip.Diagram.TurnNetwork.instSymmNodeGraph
set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

namespace CriticalStrip.Diagram
noncomputable section
open scoped Classical BigOperators
open _root_.Matrix Filter Topology

lemma physical_augmented_stationary (N : ℕ) :
    (restrictedColumn (SewnRectangle.leftWeights N)).mulVec
        (fun p=>angleAugmented (fun _ : Fin N=>lambda) p.val)+
      injectionColumn (SewnRectangle.leftWeights N)=
      fun p=>angleAugmented (fun _ : Fin N=>lambda) p.val := by
  sorry

end
end CriticalStrip.Diagram

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/NorthMass.lean (declaration OAI.CriticalStrip.Diagram.physical_augmented_stationary)

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