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.