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slug: gobel-misra-colored-path-determinant-refutation bibkey: gobel2025colored doi: 10.48550/arXiv.2506.23936 url: https://arxiv.org/abs/2506.23936v1 triage: theorem motivation_gids:

  • D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.result

Gobel-Misra Colored-Path Determinant Refutation

Problem

Gobel and Misra, arXiv:2506.23936v1, Conjecture 5.3, printed page 27, assert that two colored paths on m vertices with equal generic concentration-matrix determinants are identical, reflections, or instances of the color configuration in Theorem 3.6 when m is even and Theorem 3.8 when m is odd, allowing the listed configuration to be reflected.

The formal claim quantifies over the vertex-color type, the edge-color type, the path length, and both labelled paths. It reads the final reflection clause in the widest form: neither path, the first path, the second path, or both paths may be reflected. The repository result is Not claim.

Motivation

The frozen result D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.result gives an exact seven-vertex counterexample to the published universal claim. It compares the generic determinants in the same multivariable polynomial ring used to encode the source’s vertex and edge color constraints.

Gap

The bounded prior-resolution screen recorded in issue 8621 checked the arXiv version record, exact title and identifier searches, arXiv abstract search, and the two Semantic Scholar citations then available. It disclosed no later proof or refutation of Conjecture 5.3. Google Scholar was not verified.

Route

Use one vertex color a, two edge colors u and v, and m = 7. Give the first path edge sequence (u,v,v,u,u,v) and the second (u,u,v,u,v,v). Successive endpoint expansions of their tridiagonal concentration matrices reduce both determinants to a^7-3a^5u^2-3a^5v^2+2a^3u^4+6a^3u^2v^2+2a^3v^4-2au^4v^2-2au^2v^4.

The two paths are unequal and are not reflections. Since seven is odd, only the Theorem 3.8 branch can remain. Each of its four reflection variants fails an explicit edge-color equality, so none of the conjecture’s alternatives holds.

Falsifier

An incorrect transcription of Theorem 3.8, a mismatch between the source’s generic concentration matrix and concentration, or an error in either exact determinant expansion would invalidate the refutation. A prior published resolution of this exact conjecture would invalidate the open-problem-resolution eligibility without changing the formal counterexample.

Evidence

The formal carrier and sole public theorem are claim : Prop and result : Not claim in D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation.lean. The proof constructs both paths, proves determinant equality by pinned matrix determinant expansions and ring normalization, and rejects every conclusion alternative by exact finite evaluation. The matching Scribe theorem node binds the frozen result to this dossier with OpenProblemResolutionClaim(Refuted).

Triage

theorem; Tier 1 recent named external conjecture, preregistered in issue 8621 before proof implementation. The admission basis is open-problem-resolution; the conservative classification is proof_shape: bind-only with escape_witness: none. Its computational use is a certified-instance with a typed refutes edge from result to claim.

ASSUMED-UNVERIFIED

Literature completeness is ASSUMED-UNVERIFIED: the bounded screen cannot exclude every prior resolution. Source-to-Lean fidelity requires independent comparison with arXiv:2506.23936v1; the Lean kernel checks the formal statement and proof, not that prose correspondence. The resolution binding records the current frozen theorem and does not by itself establish publication or merge.