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

Proved

OAI.CriticalStrip.Diagram.nextScalar_homogeneous_of_ne

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.nextScalar_homogeneous_of_ne of lean/OAI/Combinatorics/StripMass/StripMass/MonicPositive.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

lemma nextScalar_homogeneous_of_ne {n : ℕ} (hn : 1≤n)
    (hp : scalarValue (RingHom.id ℂ) (fun _ : Fin n => (C:ℂ))≠0) :
    ∃ g : Polynomial ℝ, MonicProblem n g ∧
      nextScalar (RingHom.id ℂ) (fun _ : Fin n => (C:ℂ))=
        Polynomial.C (scalarValue (RingHom.id ℂ) (fun _ : Fin n => (C:ℂ))) * g.map Complex.ofRealHom

Definitions. The statement is written with OAI.CriticalStrip.Diagram.C (definitions bundle OAIQ042S000 of this split); OAI.CriticalStrip.MonicProblem (definitions bundle OAIQ042S001 of this split); OAI.CriticalStrip.Diagram.scalarValue (definitions bundle OAIQ042S007 of this split); OAI.CriticalStrip.Diagram.nextScalar (definitions bundle OAIQ042S008 of this split).

Used by. The proofs of OAI.CriticalStrip.Diagram.deformationAngles_scalar_ne_zero, OAI.CriticalStrip.Diagram.scalarValue_homogeneous_ne_zero and OAI.CriticalStrip.critical_strip_mass of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ042S007
import Definitions.Def_OAIQ042S008

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.Diagram
noncomputable section
open Filter Polynomial
open scoped Topology BigOperators Classical

lemma nextScalar_homogeneous_of_ne {n : ℕ} (hn : 1≤n)
    (hp : scalarValue (RingHom.id ℂ) (fun _ : Fin n => (C:ℂ))≠0) :
    ∃ g : Polynomial ℝ, MonicProblem n g ∧
      nextScalar (RingHom.id ℂ) (fun _ : Fin n => (C:ℂ))=
        Polynomial.C (scalarValue (RingHom.id ℂ) (fun _ : Fin n => (C:ℂ))) * g.map Complex.ofRealHom := 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/MonicPositive.lean (declaration OAI.CriticalStrip.Diagram.nextScalar_homogeneous_of_ne)

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