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Abstract

We give a complete rational reconstruction of an explicit obstruction to positive definiteness for a fixed-center product of the Riemann theta density. At s_0=(1)/(4)log(13/(4π)), the signed derivative functional evaluated on the square of P(X)=X^5-264X^3+10952X lies in the exact rational interval [-1412370791.837791749, -1412370791.837791748]. The proof constructs derivatives by formal power series, retains twelve theta summands, bounds the entire remaining series, and encloses exponentials by range-reduced alternating series. A printed table of rational moment intervals proves the negative contraction. The odd derivative Hankel block is indefinite, and the six-point translation Gram has a negative direction at every sufficiently small positive spacing.