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

Proved

OAI.Zarankiewicz.TruncatedSpokeCoordinates.exists_cycle_block_radius

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.exists_cycle_block_radius of lean/OAI/Combinatorics/Crossing/CycleBlockIntersections.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Zarankiewicz.TruncatedSpokeCoordinates):

theorem exists_cycle_block_radius (hnorm : NoAdjacentCrossings D)
    (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D) :
    ∃ δ : ℝ, 0 < δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε| < δ →
      ∀ (d f : ℕ → (completeBipartite m n).Dart)
        (hd : ∀ i, (d i).snd=(d (i+1)).fst) (hf : ∀ j, (f j).snd=(f (j+1)).fst) (i j : ℕ),
      bipartiteDartEdge (d (i+1)) ≠ bipartiteDartEdge (d i) →
      bipartiteDartEdge (f (j+1)) ≠ bipartiteDartEdge (f j) →
      (V.dartCycleBlock d hd i).Transverse (C.shiftedCycleBlock f hf ε j) ∧
      polygonIntersection (V.dartCycleBlock d hd i).vertices (C.shiftedCycleBlock f hf ε j).vertices
        (V.dartCycleBlock d hd i).pieces (C.shiftedCycleBlock f hf ε j).pieces=
          bipartiteDartSign (d i)*bipartiteDartSign (f j)*
            V.signedIntersection (bipartiteDartEdge (d i)) (bipartiteDartEdge (f j))+
              C.passageIntersection d f hd hf i j ε

Definitions. The statement is written with OAI.Zarankiewicz.PolygonalChain (definitions bundle OAIB3Zr000 of this split); OAI.Zarankiewicz.NoAdjacentCrossings, OAI.Zarankiewicz.bipartiteDartEdge, OAI.Zarankiewicz.bipartiteDartSign and OAI.Zarankiewicz.completeBipartite (definitions bundle OAIB3Zr001 of this split); OAI.Zarankiewicz.LocallyStraightCrossings, OAI.Zarankiewicz.LocallyTransverseCrossings, OAI.Zarankiewicz.TruncatedSpokeCoordinates, OAI.Zarankiewicz.VertexTruncation, OAI.Zarankiewicz.polygonIntersection, OAI.Zarankiewicz.PolygonalChain.Transverse, OAI.Zarankiewicz.TruncatedSpokeCoordinates.passageIntersection, OAI.Zarankiewicz.TruncatedSpokeCoordinates.shiftedCycleBlock, OAI.Zarankiewicz.TruncatedSpokeCoordinates.shiftedCycleVertex, OAI.Zarankiewicz.VertexTruncation.dartCycleBlock, OAI.Zarankiewicz.VertexTruncation.dartCycleVertex and OAI.Zarankiewicz.VertexTruncation.signedIntersection (definitions bundle OAIB3Zr002 of this split); OAI.Zarankiewicz.AdmissibleDrawing, OAI.Zarankiewicz.BipartiteEdge and OAI.Zarankiewicz.Plane (the published bundle Def_BipartiteCrossing).

Used by. The proofs of OAI_Zarankiewicz_TruncatedSpokeCoordinates_closed_cycle_matrix_sum and OAI.Zarankiewicz.TruncatedSpokeCoordinates.exists_closed_cycle_intersection_radius of this split import 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

namespace TruncatedSpokeCoordinates
variable {m n : ℕ} {D : AdmissibleDrawing m n} {V : VertexTruncation D.toContinuousDrawing}
  (C : TruncatedSpokeCoordinates V)

theorem exists_cycle_block_radius (hnorm : NoAdjacentCrossings D)
    (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D) :
    ∃ δ : ℝ, 0 < δ ∧ ∀ ε : ℝ, ε ≠ 0 → |ε| < δ →
      ∀ (d f : ℕ → (completeBipartite m n).Dart)
        (hd : ∀ i, (d i).snd=(d (i+1)).fst) (hf : ∀ j, (f j).snd=(f (j+1)).fst) (i j : ℕ),
      bipartiteDartEdge (d (i+1)) ≠ bipartiteDartEdge (d i) →
      bipartiteDartEdge (f (j+1)) ≠ bipartiteDartEdge (f j) →
      (V.dartCycleBlock d hd i).Transverse (C.shiftedCycleBlock f hf ε j) ∧
      polygonIntersection (V.dartCycleBlock d hd i).vertices (C.shiftedCycleBlock f hf ε j).vertices
        (V.dartCycleBlock d hd i).pieces (C.shiftedCycleBlock f hf ε j).pieces=
          bipartiteDartSign (d i)*bipartiteDartSign (f j)*
            V.signedIntersection (bipartiteDartEdge (d i)) (bipartiteDartEdge (f j))+
              C.passageIntersection d f hd hf i j ε := by
  sorry

end TruncatedSpokeCoordinates
end Zarankiewicz

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/CycleBlockIntersections.lean (declaration OAI.Zarankiewicz.TruncatedSpokeCoordinates.exists_cycle_block_radius)

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