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

The coefficient-parity conjecture of Forsgård and Shapiro uses the indices at which a_k^2 - a_{k-1}a_{k+1} is nonnegative to bound the number of real zeros of a polynomial with positive coefficients. We give an explicit degree-77 polynomial with positive rational coefficients for which the selected indices 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 required root crossing; all coefficients and signs can also be checked by rational arithmetic. This gives a complete counterexample to Conjecture 10 in Section 7 of Boris Shapiro, Problems Around Polynomials: The Good, The Bad and The Ugly... (2015), DOI 10.1007/s40598-015-0008-4; Hugging Face dataset identifier AMR-021-0015. It does not settle neighboring Conjecture 9. The uploaded package includes the English manuscript, source files, and exact-arithmetic reproduction. This is an unrefereed preprint; no minimal-degree, exact-root-count, or absolute-priority claim is made. AI-assisted tools were used in exploratory algebra, source checking, manuscript preparation, and independent exact-arithmetic checks; the author is responsible for the arguments.