OAI.Zarankiewicz.TruncatedSpokeCoordinates.odd_normalized_crossing_lower_bound: intermediate lemma from OpenAI, The crossing number of complete bipartite graphs

Proved

OAI.Zarankiewicz.TruncatedSpokeCoordinates.odd_normalized_crossing_lower_bound

by dbenbenn · Oct 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Zarankiewicz.mainTarget_proof, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, The crossing number of complete bipartite graphs, OpenAI Math Release, September 23, 2026, result family 165, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.Zarankiewicz.TruncatedSpokeCoordinates.odd_normalized_crossing_lower_bound of lean/OAI/Combinatorics/Crossing/OddNormalizedLowerBound.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Zarankiewicz.TruncatedSpokeCoordinates, with open scoped BigOperators):

theorem odd_normalized_crossing_lower_bound (C : TruncatedSpokeCoordinates V) (hnorm : NoAdjacentCrossings D)
    (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D)
    (s : ℕ) (hs : 0 < s) (hcard : n=2*s+1) (hm : 0 < m) :
    axisPairs m*(s*s) ≤ crossingCount D

Definitions. The statement is written with OAI.Zarankiewicz.NoAdjacentCrossings (definitions bundle OAIB3Zr001 of this split); OAI.Zarankiewicz.LocallyStraightCrossings, OAI.Zarankiewicz.LocallyTransverseCrossings, OAI.Zarankiewicz.TruncatedSpokeCoordinates and OAI.Zarankiewicz.VertexTruncation (definitions bundle OAIB3Zr002 of this split); OAI.Zarankiewicz.AdmissibleDrawing, OAI.Zarankiewicz.axisPairs and OAI.Zarankiewicz.crossingCount (the published bundle Def_BipartiteCrossing).

Used by. The proof of OAI.Zarankiewicz.mainTarget_proof of this split imports it.

Preamble

import Mathlib
import Lean.Elab.Tactic.Omega
import Definitions.Def_BipartiteCrossing
import Definitions.Def_OAIB3Zr002

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 Zarankiewicz.TruncatedSpokeCoordinates
open scoped BigOperators
variable {m n : ℕ} {D : AdmissibleDrawing m n} {V : VertexTruncation D.toContinuousDrawing}
  (C : TruncatedSpokeCoordinates V)

theorem odd_normalized_crossing_lower_bound (C : TruncatedSpokeCoordinates V) (hnorm : NoAdjacentCrossings D)
    (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D)
    (s : ℕ) (hs : 0 < s) (hcard : n=2*s+1) (hm : 0 < m) :
    axisPairs m*(s*s) ≤ crossingCount D := by
  sorry

end Zarankiewicz.TruncatedSpokeCoordinates

end

end OAI
end

Source

OpenAI, The crossing number of complete bipartite graphs, OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 165; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-crossing-number-of-complete-bipartite-graphs-September-23-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Combinatorics/Crossing/OddNormalizedLowerBound.lean (declaration OAI.Zarankiewicz.TruncatedSpokeCoordinates.odd_normalized_crossing_lower_bound)

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