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Abstract

We classify the least degree of a polynomial whose square has negative signed-moment value for the normalized fixed-center Riemann theta product. On 3≤ x≤12 the degree has fourteen transitions, takes values between five and fifteen, and is nonmonotone. We isolate every transition and determine the degree at equality. Jacobi symmetry extends the complete classification to π^2/12≤ x≤12. We quantify how the two- and three-summand theta models shift all fourteen crossings: the respective root errors are below 0.00049 and 2.588 × 10^(-10). A constructive finite-difference bound converts polynomial obstructions into finite translation Gram witnesses. Twenty explicit rational certificates include a seven-point example stable under absolute entrywise errors of 1.94 × 10^(-7). We also prove fixed-degree Gaussian limits and single-summand degree-five positivity. Infinite tails, directed interval covers and independent rational checks are included in a self-contained reproduction supplement.