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Abstract

Monotone center averaging makes the complete Riemann theta product moment matrix positive definite through degree eight, under the corresponding moment-integrability hypothesis. The lower-degree conclusions retain their weaker hypotheses, and every finite physical parameter is included. The proof separates positive mean Schur residuals from variance reserves, with complete-theta tail and finite-interval certificates. This finite positivity has a quantitative extension but an all-order boundary. At every fixed degree N, the local moment Gram has a 99/100 reserve in normalized Hermite coordinates for X≥1000N^2. In contrast, every sufficiently large center cutoff admits negative rational polynomial squares in both parities, of degree at most ⌈ A^(31/4)⌉. At A=23/2, explicit witnesses have degrees 4096 and 4097, with certified complete-theta moment signs. Positive, strictly increasing bounded real-analytic center weights also yield negative polynomial squares. The local physical consequences include fixed configurations of at most nine points. Classical Schur, approximation and quadrature tools are identified as inputs; the complete-theta estimates, witnesses and quantitative domains are the results studied here.