A Reciprocal-Block Counterexample to the Forsgård–Shapiro Coefficient-Parity Bound

Version 1.1 corrects a missed direct prior reference: Katkova, Shapiro and Vishnyakova already disproved this conjecture in 2024 (In search of Newton-type inequalities, Section 5; DOI 10.1016/j.jmaa.2024.128349; arXiv:2403.12200). We give an alternative degree-77 polynomial with positive rational coefficients for which all indices with a_k^2 - a_(k-1)a_(k+1) >= 0 change parity only once, but which has at least three distinct negative real zeros. The construction joins a degree-38 block to its reciprocal; a weighted sum-of-squares identity proves the root crossing. The original proof and exact rational checks are unchanged. The 2024 counterexample also yields a distinct-root violation by a strict-margin perturbation. This is not a first disproof or a new problem closure; no construction-priority or minimum-degree claim is made. The neighboring weighted conjecture is already false as well, although the unweighted witness cannot be reused unchanged. Corpus identifier AMR-021-0015. English, AI-assisted, self-audited, unrefereed preprint. No independent human review, formal proof-assistant certification or exact-total-root-count claim. Zenodo publication is not peer review. The author remains responsible for the final text. Paper page: https://eulersolve.org/papers/amr-021-0015/