slug: suvagiya-conjecture28-refutation
bibkey: suvagiya2026parity
doi: 10.48550/arXiv.2607.17343
url: https://arxiv.org/html/2607.17343v2
triage: theorem
motivation_gids:
  - D5/S3/Combinatorics/Graph/SuvagiyaSignedSquareCycleRefutation.result

Suvagiya Conjecture 28 refutation

Problem

Conjecture 28 in section 12.3 of arXiv:2607.17343v2 states that for every integer m >= 4, the minimum spectral radius over all independent edge signings of C_(8m)(1,2) equals the greatest real root of x^4−2x^3−6x^2+12x−4. The radius is the attained maximum absolute eigenvalue of the real symmetric adjacency matrix. Both Hamilton-cycle sign products are included.

Motivation

The theorem D5/S3/Combinatorics/Graph/SuvagiyaSignedSquareCycleRefutation.result negates that complete assertion. The concrete counterexample has 32 vertices and does not belong to the periodic gauge slice supplying Theorem 26’s upper bound.

Gap

The exact unrestricted optimum at n = 32 and at other sizes is not determined. No repaired restriction on signings is proved.

Route

The step-one signs are +1 except a_31 = −1. The step-two signs repeat (1,1,−1,1,−1,−1,1,−1), with b_30 = −1 and b_31 = +1. Exact finite Horner identities show R(A^2) = 0, where R(y) = y^8−32y^7+416y^6−2816y^5+10568y^4−21632y^3+22168y^2−9408y+1262. All coefficients of R((279/100)^2+z) are positive. Spectral mapping therefore bounds the attained radius by 279/100. The quartic has a root between 279/100 and 3, since its endpoint values are −6766519/100000000 and 5. Its greatest root is strictly larger than the radius of this signing.

Falsifier

A discrepancy in any seam edge, failure of a finite Horner identity, a nonpositive coefficient in the shifted polynomial, or a source restriction excluding independent signings with Hamilton sign product −1 would invalidate the proposed counterexample to this assertion.

Evidence

The formal source defines the complete finite signing carrier, actual symmetric adjacency, attained spectral radii, quartic roots, and closed integer-indexed claim. Its sole public theorem is result : Not claim. The mathematical statement is the unrestricted source conjecture, including its attained extrema and complete sign domain.

The escape audit is unfinished ([issue #11789](https://github.com/the-omega-institute/trureturing/issues/11789)). The faithful Reg mirror compiles, but registration processing reports IE-C050 (E2.unknown_constant at LE.mk). No declared_validated registration or accepted source-binding evidence is established.

Triage

The unrestricted lower bound is refuted; Theorem 26’s upper bound is unaffected. This is a Tier 1 named external conjecture, preregistered in issue 11744. The admission basis is open-problem-resolution, the proof shape is bind-only, and the escape witness is none. Its computational use is certified-instance with refutes from result to claim.

ASSUMED-UNVERIFIED

Exhaustive literature completeness and publication priority are unverified. The Lean kernel verifies the formal proposition and proof; independent review checks correspondence with the source’s definitions and quantifiers.