A Positive-Coefficient Counterexample to the Weighted Forsgård–Shapiro 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 present an alternative explicit degree-1,000,077 positive-rational-coefficient polynomial with one parity change among all indices where (k+1)a_k^2 - k a_(k-1)a_(k+1) > 0, but at least three distinct negative real zeros. An even shift of a degree-77 reciprocal-block seed and a tiny strictly log-convex factorial prefix give a compact exact formula. The mathematical construction is unchanged. The 2024 family also violates the distinct-root reading after a strict-margin perturbation. This is not a first disproof or a new problem closure; no construction-priority or optimal-degree claim is made. Corpus identifier AMR-021-0014. 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-021-0014/