A Certified Obstruction to the Budget-12 Positive-Part Criterion on an Origin-Containing Interval for Weil Positivity at Support Three

This research letter establishes certified obstructions to twodistinct Positive-Part sufficiency tests for Weil positivity atFourier support a = 3, and demonstrates the complementary strengthof the Zero-Gram method. First, using the actual arithmetic phases of the prime-power terms,we construct an explicit 50-term odd Paley–Wiener trial functionwith exact rational coefficients. Rigorous Arb interval integrationover 65 subintervals proves that the norm of the budget-12Positive-Part compression restricted to [-60, 60] exceeds12.002190467312. Consequently, the global budget-12 Positive-Partsufficient condition fails even in the odd sector. This obstruction concerns an interval containing the origin, notthe point t = 0 itself, and does not imply negativity of thesigned Weil quadratic form. Second, for the same trial function, a separate Zero-Gramcertificate establishes strict positivity of the signed Weilquadratic form, with normalized value exceeding6.14948083 × 10^-6. The certificate uses the first 1,000nontrivial zeros of the Riemann zeta function, the establishedverification of the Riemann hypothesis up to height3 × 10^12, and an explicit bound for the remaining zerocontributions. Thus, a direction for which the global Positive-Partcriterion fails is nevertheless certified positive by thezero representation. This provides a rigorous, explicitexample of complementarity between the two methods. Third, an independent Arb certificate establishes aphase-uniform obstruction for the freely varying prime-phasemodel. At the physical frequency T = 1,369,048.1155,a normalized reproducing-kernel trial function gives thecertified lower bound R_5(3, C) > 29.6497032216 > 12. By monotonicity, the specified phase-uniform thresholdtest fails for every 0 < T <= 1,369,048.1155. This third result is mathematically distinct from theactual-phase obstruction: the phase-uniform model allowsindependent prime phases and does not require the obstructingphase configuration to occur along the actual arithmetic orbit. The accompanying reproducibility archive includes the exacttrial-function coefficients, Arb certificate generators andverifiers, machine-readable JSON certificates, execution logs,and supporting numerical diagnostics. The rigorous computationsuse python-flint 0.9.0, while the floating-point diagnosticsuse NumPy 2.5.3 and SciPy 1.18.1. Together, these results identify precise limitations ofparticular Positive-Part sufficient conditions and providea concrete motivation for retaining signed contributionsthrough complementary Zero-Gram and Positive-Part methods. The letter does not prove the Riemann hypothesis, establishnegativity of the Weil quadratic form, or rule out otherPositive-Part bounds or hybrid positivity methods.