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slug: pandey-parity-conjecture-refutation bibkey: pandey2026parity doi: null url: https://arxiv.org/abs/2601.03293v1 triage: theorem motivation_gids:

  • D5/S3/StatisticalMechanics/HardCore/IndependentPartitionDeletion.independencePolynomial_eval

Pandey’s Parity Conjecture 4.1

Problem

Rohan Pandey, arXiv:2601.03293v1, Conjecture 4.1 (Parity Conjecture), PDF page 4:

For all integers n ≥ 2k+1, the independence polynomial I(GP(n,k),x) has only real roots if and only if k is even.

Definition 2.1 supplies n >= 3 and 1 <= k < n/2. Precisely, for every n,k : Nat with 3 <= n, 1 <= k, and 2*k < n, the assertion is (forall z : Complex, eval₂ (Int.castRingHom Complex) z (independencePolynomial (gp n k) univ) = 0 -> z.im = 0) <-> Even k. The polynomial is the existing integer independent-set generating polynomial on all vertices. No experimental-range restriction is added.

Motivation

The frozen IndependentPartitionDeletion.independencePolynomial_eval relates this actual polynomial to its weighted independent-set sum. It allows an exact graph counterexample to address the published biconditional without assuming a transfer-matrix formula or approximate root calculations.

Gap

There is no remaining gap in the resolution of this exact conjecture. The earlier public refutation already gives GP(3,1) with polynomial 1+6x+6x^2 and an explicit opposite-parity isomorphism GP(7,2) -> GP(7,3) that contradicts the full biconditional. Its provider commit timestamp is 2026-06-13T01:23:03Z; its internal June 11 date is unverified. The prior-refutation Library note pins the commit, blob, and source hash and explains the edge map. This evidence supersedes the bounded no-hit screen for issue 8619 and invalidates the historical open-problem-resolution novelty admission. The local contribution is a formalization of a known refutation, not an eligible newly solved open problem.

Route

Use vertices Bool x Fin n, representing u_i and v_i, and the undirected, loop-free closure of the source’s outer-successor, inner-k-step, and equal-index spoke relations, with indices modulo n. At n=3,k=1, all source hypotheses hold. The independent sets are exactly the empty set, the six singletons, and the six cross-layer pairs with different indices. Their generating polynomial is 1+6X+6X^2.

For a complex zero z=a+bi, the imaginary equation is 6b(1+2a)=0. If b != 0, then a=-1/2, and the real equation becomes -1/2-6b^2=0, impossible since b^2 >= 0. Thus every complex root is real, whereas Even 1 is false. This refutes the forward implication and hence the full universal biconditional.

Falsifier

A source restriction excluding n=3,k=1, an incorrect graph-to-source mapping, or a failure of the exact independent-set enumeration would invalidate this refutation. The prior-resolution falsifier of novelty admission is satisfied by the exact public refutation cited above; that correction does not invalidate the mathematical counterexample.

Evidence

The formal carrier and sole public result result : Not claim are in D5/S3/Combinatorics/GeneralizedPetersen/ParityRefutation.lean. The proof uses kernel decide for the configuration equality, the frozen evaluation bridge, finite-sum simplification, and exact complex arithmetic. The matching Scribe definition attributes the result to the earlier public refutation through its separate Library note and retains OpenProblemResolutionClaim(Refuted). Under spec §11.20.5 this records the local theorem-to-problem binding, not worldwide novelty or the validity of historical novelty admission. The graph and claim retain the Pandey Library source.

Primary-source hashes: TeX 5147f84b91867223e59aa63486af9b4276eadd08dc785fa40ab3c670c80ef004; PDF bae13d4946674f1149b74ef9095d04d9d7ec2d4010dbd8357de52ad88cd066e2.

Triage

theorem; a published named conjecture with a known prior exact refutation. Its historical Tier 1 selection in issue 8619 did not establish eligibility: the open-problem-resolution admission basis was invalid because the exact assertion was already resolved. The conservative proof_shape: bind-only and escape_witness: none remain, without a replacement admission basis or escape-witness retrofit. The existing valid frozen mathematics is retained under CLAUDE §§1.3 and 3.2. The computational use remains a certified-instance refuting the closed full claim. The triangular prism, its polynomial, and its real-rootedness are classical, including the claw-free theorem of Chudnovsky and Seymour cited by the source. No new family, technique, classification, or global priority is claimed.

ASSUMED-UNVERIFIED

The public note’s internal June 11 date and its numerical, enumeration, Sturm, checker, and audit claims are unverified here; none is needed for the explicit isomorphism contradiction or the prior-art correction. Literature completeness beyond this exact hit is ASSUMED-UNVERIFIED; no earliest-priority claim is made. Source-to-Lean fidelity requires independent comparison with the cited version; the kernel checks the formal statement and proof, not that prose correspondence. The typed binding establishes neither worldwide novelty nor publication.