Refutation of the panchromatic pairing conjecture

Lalou, Mbarek, Skender, and Togni, Completely Independent Spanning Trees in Split Graphs: Structural Properties and Complexity, arXiv:2512.15486v2, Section 5, Conjecture 1 (verbatim apart from plain-text typography):

Every hypergraph H satisfies chi_p^2(H) = chi_p(H) - ceil(alpha_{chi_p(H)}(H)/2).

Here chi_p and chi_p^2 are the maximum feasible panchromatic and bipanchromatic color counts. A bipanchromatic coloring requires every global color class to occur at least twice. The quantity alpha_k is the minimum number of globally unique colors over all panchromatic k-colorings.

The published equality has a six-vertex counterexample with three four-element edges and no isolated vertex. Formalizing the actual maxima and the minimum over all maximizing colorings tests the precise extremal content rather than a bounded surrogate or an assumed invariant value.

The exact source statement, six-vertex witness, success and falsification criteria, and bounded literature-status scope were preregistered in issue #8518 before the first Lean edit. A bounded screen of the primary and latest arXiv records, arXiv results for bipanchromatic coloring, author and OpenAlex records, the project library, pinned Mathlib, and public Lean keyword results found no exact earlier resolution in the searched scope. Crossref records the journal publication as Discrete Applied Mathematics 391 (2026), pages 511--522, DOI 10.1016/j.dam.2026.05.028; publisher full text remains unverified because access returned HTTP 403. No worldwide novelty or priority claim is made.

Let the vertex type be Fin 6 and take edges {0,1,2,3}, {0,1,2,4}, and {0,1,2,5}. Coloring the core by three distinct colors and all petals by a fourth proves panchromatic feasibility at four. Any panchromatic coloring maps the first four-element edge onto all colors, so its color count is at most four.

For an arbitrary panchromatic four-coloring, the first edge map is injective. The other two edges force vertices 4 and 5 to have vertex 3's missing color. Thus the three core color fibers are singletons in every maximizing coloring; the displayed coloring attains exactly three singleton colors. Hence alpha_4 is three.

Color vertex v by v modulo three to obtain a bipanchromatic three-coloring. For any bipanchromatic k-coloring, the k global fibers each have cardinality at least two and their cardinalities sum to six, so k is at most three. Therefore the extrema are chi_p=4, alpha_4=3, and chi_p^2=3, while the conjectured right side is two.

A defect in the refutation would be exposed by a panchromatic coloring using more than four colors, a panchromatic four-coloring with fewer than three singleton global fibers, or a bipanchromatic coloring using more than three colors. Any mismatch between the finite-set definitions and the source's global color multiplicities would also invalidate the result.

The 207-line Lean module D5/S3/ConceptDynamics/GraphColoring/PanchromaticPairingConjectureRefutation has exactly the public surface claim : Prop and result : Not claim. The claim uses relational extrema with attainment and universal inequalities; the result constructs all three witness certificates. Both declarations use only the standard axiom closure [propext, Classical.choice, Quot.sound]. The finite witness checks use kernel decide, not native_decide.

theorem; resolution refuted. The public result has proof_shape: content: its live proof constructs the source model, both universal maximum bounds, and the all-optimal-coloring singleton lower bound. The public conclusion itself is the escape witness. Its admission_basis is open-problem-resolution, and its computational utility is the typed certified-instance refutation from result to claim.

The bounded source and literature search is not exhaustive coverage and does not establish worldwide novelty or priority. The journal full text and any unpublished answers to the conjecture remain unverified.