bibkey: demonstrandum2026pandeyrefutation authors: demonstrandum-research/artifacts (repository publisher) year: 2026 title: “Refutation of the Parity Conjecture for Independence Polynomials of Generalized Petersen Graphs (arXiv:2601.03293, Conjecture 4.1)” doi: null url: https://github.com/demonstrandum-research/artifacts/blob/94db9ed50d48a57aae5ccb72e6a95a2b8f8f39d3/problems/p2-factory/kills/pandey-parity/WRITEUP.md claim: “Prior full refutation of Pandey’s Conjecture 4.1: GP(3,1) has real-rooted independence polynomial 1+6x+6x^2 with odd step, and the explicit isomorphism GP(7,2) to GP(7,3) contradicts the universal parity biconditional.” strata_touched:

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

Prior public refutation of Pandey’s Parity Conjecture

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Exact prior refutation

The graph domain and conjecture are recorded in the separate Pandey Library note: n >= 3, 1 <= k, 2*k < n, with no restriction to the experimental range 20 <= n <= 30. Section 3(c) of the prior note explicitly gives the triangular prism GP(3,1), its polynomial 1+6x+6x^2, and its real-rootedness although k=1 is odd. This is the exact counterexample formalized locally.

Section 3(d) gives GP(7,2) -> GP(7,3) by u_j -> v'_(3j mod 7), v_j -> u'_(3j mod 7). Multiplication by 3 is invertible modulo 7. Outer edges map to step-3 inner edges, step-2 inner edges map to outer edges with step 6 = -1 mod 7, and spokes map to spokes. Both parameter pairs satisfy the source domain. Thus the graphs are isomorphic and have equal independent-set counts and polynomials, while their steps have opposite parity. The full universal biconditional would assign contradictory real-rootedness predictions to this same polynomial. This argument needs no computed roots and already refutes the full assertion. The source’s additional numerical, enumeration, Sturm, checker, and audit claims are not verification evidence adopted here.

Local formalization and attribution

The problem dossier and canonical Blueprint record the local Lean formalization of the known GP(3,1) refutation, not a formalization of the source’s other counterexamples or isomorphism. The result’s structured literature citation refers to this note; the graph and conjecture retain Pandey provenance.

The prior exact resolution supersedes the bounded no-hit literature screen for issue 8619 and invalidates the historical open-problem-resolution novelty-admission basis. This is not an eligible newly solved open problem. The conservative proof_shape: bind-only assessment remains; no replacement admission basis or escape witness is asserted. The valid frozen mathematics is retained under CLAUDE §§1.3 and 3.2. Under spec §11.20.5, the typed Refuted record binds the local theorem to the problem without claiming worldwide novelty. Literature completeness beyond this exact hit is ASSUMED-UNVERIFIED; no earliest-priority claim is made.