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

Proved

OAI.Zarankiewicz.VertexTruncation.exists_all_edge_shift_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.VertexTruncation.exists_all_edge_shift_radius of lean/OAI/Combinatorics/Crossing/UniformTrackPerturbation.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

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

theorem exists_all_edge_shift_radius (V : VertexTruncation D.toContinuousDrawing)
    (hnorm : NoAdjacentCrossings D) (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D) :
    ∃ δ : ℝ, 0 < δ ∧ ∀ t : ℝ, t ≠ 0 → |t| < δ → ∀ e f : BipartiteEdge m n,
      ∀ i : Fin (V.pieces e), ∀ j : Fin (V.pieces f),
        SegmentPairTransverse (V.vertices e i.val) (V.vertices e (i.val+1))
          (shiftedTrack (V.vertices f) (V.pieces f) t j.val)
          (shiftedTrack (V.vertices f) (V.pieces f) t (j.val+1)) ∧
        segmentCrossingSign (V.vertices e i.val) (V.vertices e (i.val+1))
          (shiftedTrack (V.vertices f) (V.pieces f) t j.val)
          (shiftedTrack (V.vertices f) (V.pieces f) t (j.val+1)) =
          if e=f then 0 else segmentCrossingSign (V.vertices e i.val) (V.vertices e (i.val+1))
            (V.vertices f j.val) (V.vertices f (j.val+1))

Definitions. The statement is written with OAI.Zarankiewicz.SegmentPairTransverse (definitions bundle OAIB3Zr000 of this split); OAI.Zarankiewicz.NoAdjacentCrossings (definitions bundle OAIB3Zr001 of this split); OAI.Zarankiewicz.LocallyStraightCrossings, OAI.Zarankiewicz.LocallyTransverseCrossings, OAI.Zarankiewicz.VertexTruncation, OAI.Zarankiewicz.segmentCrossingSign, OAI.Zarankiewicz.shiftedTrack, OAI.Zarankiewicz.VertexTruncation.pieces and OAI.Zarankiewicz.VertexTruncation.vertices (definitions bundle OAIB3Zr002 of this split); OAI.Zarankiewicz.AdmissibleDrawing and OAI.Zarankiewicz.BipartiteEdge (the published bundle Def_BipartiteCrossing).

Used by. The proofs of OAI.Zarankiewicz.TruncatedSpokeCoordinates.exists_cycle_block_radius and OAI.Zarankiewicz.VertexTruncation.exists_uniform_track_intersections 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.VertexTruncation
open scoped BigOperators
variable {m n : ℕ} {D : AdmissibleDrawing m n}

theorem exists_all_edge_shift_radius (V : VertexTruncation D.toContinuousDrawing)
    (hnorm : NoAdjacentCrossings D) (hstraight : LocallyStraightCrossings D) (htrans : LocallyTransverseCrossings D) :
    ∃ δ : ℝ, 0 < δ ∧ ∀ t : ℝ, t ≠ 0 → |t| < δ → ∀ e f : BipartiteEdge m n,
      ∀ i : Fin (V.pieces e), ∀ j : Fin (V.pieces f),
        SegmentPairTransverse (V.vertices e i.val) (V.vertices e (i.val+1))
          (shiftedTrack (V.vertices f) (V.pieces f) t j.val)
          (shiftedTrack (V.vertices f) (V.pieces f) t (j.val+1)) ∧
        segmentCrossingSign (V.vertices e i.val) (V.vertices e (i.val+1))
          (shiftedTrack (V.vertices f) (V.pieces f) t j.val)
          (shiftedTrack (V.vertices f) (V.pieces f) t (j.val+1)) =
          if e=f then 0 else segmentCrossingSign (V.vertices e i.val) (V.vertices e (i.val+1))
            (V.vertices f j.val) (V.vertices f (j.val+1)) := by
  sorry

end Zarankiewicz.VertexTruncation

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/UniformTrackPerturbation.lean (declaration OAI.Zarankiewicz.VertexTruncation.exists_all_edge_shift_radius)

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