A Positive-Coefficient Counterexample to the Weighted Forsgård–Shapiro Parity Bound

Forsgård and Shapiro proposed bounding the number of real zeros of a positive-coefficient polynomial by the parity changes among all indices at which (k+1)a_k^2 - k a_(k-1)a_(k+1) is positive. We give a counterexample with positive rational coefficients, parity count one, and at least three distinct negative real zeros. An even-degree shift of a degree-77 reciprocal-block seed makes the undesired weighted local quantities negative. A sufficiently small, strictly log-convex factorial prefix fills every missing coefficient without introducing a parity change or destroying the three roots. The explicit witness has degree 1,000,077; no optimal-degree claim is made. Its coefficients have a compact exact formula.