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Weighted Bond Difference Counterexample

Abstract

Three disjoint triangles and three spanning paths refute Conjecture 4.13(2).

Definition 1.1 (The literal all-graphs conjecture).

Formalization. D5/S0/Certificates/GonzalezDLeonWachsWeightedBondDifferenceRefutation.claim (✓ std3).

Citation. Rafael S. Gonzalez D’Leon; Michelle L. Wachs (2026). Weighted bond posets and a new chromatic symmetric function. DOI: 10.48550/arXiv.2608.08692. URL: https://arxiv.org/html/2608.08692v1#S4.Thmtheorem13.

Commentary.

For every finite graph G and spanning subgraph H with the same number of connected components, the sign-corrected difference of their literal source Mobius polynomials is asserted to split over the reals. This is part (2) as written; no connected-only premise is added.

Theorem 1.2 (The nine-vertex witness refutes part (2)).

Proof. Machine-checked in Lean as D5/S0/Certificates/GonzalezDLeonWachsWeightedBondDifferenceRefutation.result (✓ std3). ∎

Resolves. Problems/gonzalez-dleon-wachs-conjecture-4-13-2-refutation (refuted) by D5/S0/Certificates/GonzalezDLeonWachsWeightedBondDifferenceRefutation.result.

Source. Repository-derived.

Acknowledgement. Rafael S. Gonzalez D’Leon; Michelle L. Wachs (2026). Weighted bond posets and a new chromatic symmetric function. DOI: 10.48550/arXiv.2608.08692. URL: https://arxiv.org/html/2608.08692v1#S4.Thmtheorem13.

Commentary.

On Fin 3 x Fin 3, G is three triangle fibers and H is three path fibers. Both have three components and H is proper. Restriction and gluing give the source-poset product, hence polynomials A^3 and B^3. Their difference is (X+1)^2 times a quartic with no real root, so it does not split.

References