bibkey: gobel2025colored authors: Hannah Göbel, Pratik Misra year: 2025 title: Linear relations of colored Gaussian cycles doi: 10.48550/arXiv.2506.23936 url: https://arxiv.org/abs/2506.23936v1 claim: Conjecture 5.3 asserts that determinant-equal colored paths differ only by the listed symmetries and configurations. strata_touched:

• D5/S3/Combinatorics/GobelMisraColoredPathDeterminantRefutation license: citation-only triage: anchor

Linear Relations of Colored Gaussian Cycles

Conjecture 5.3 on printed page 27 asserts that two colored paths with equal generic concentration-matrix determinants are identical, reflections, or instances of the configurations in Theorem 3.6 for even order and Theorem 3.8 for odd order. The paper uses disjoint vertex- and edge-color sets and treats color labels as part of a colored path.

Theorem 3.6 on printed page 13 gives the four even-order clauses. Theorem 3.8 on printed page 15 gives the three odd-order clauses. Its third clause prints edge indices through m, although the final indexed edge would require a nonexistent vertex m+1; the formal statement uses the actual edge positions 1 through m-1.

This note attests the definitions, configurations, and published conjecture, not the repository’s refutation. The arXiv version record, exact-title and identifier searches, arXiv abstract search, and the two Semantic Scholar citations available during verification disclosed no later settlement. Google Scholar was not verified.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2506.23936
• URL: https://arxiv.org/abs/2506.23936v1
• Version and location: arXiv:2506.23936v1, printed pages 4-5, 13, 15, and 27.