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 a rigorous, computer-assisted obstructionto the budget-12 positive-part criterion for Weil positivity at Fouriersupport a = 3. Using the actual arithmetic phases of the prime-power terms, rather thanindependently varied phases, we construct an explicit 50-term oddPaley–Wiener trial function with rational coefficients. Certified Arbinterval integration over 65 subintervals yields the strict operatornorm lower bound ||H_{3,12,60}^-|| > 12.002190467312 > 12. Thus, the global budget-12 positive-part sufficient condition failsalready on the origin-containing interval [-60, 60]. The obstructiondoes not imply negativity of the signed Weil quadratic form, nor doesit identify the point t = 0 as the cause of the failure. A second result demonstrates the complementarity of the tworepresentations. For the same trial function, a zero-side certificateusing the first 1,000 nontrivial zeta zeros, the established verificationof the Riemann hypothesis up to height 3 × 10^12, and an explicitzero-tail estimate proves that the signed Weil quadratic form isstrictly positive, with normalized value exceeding 6.149 × 10^-6. Consequently, the positive-part criterion fails to establish positivityalong a direction for which the zero representation provides arigorous positive lower bound. The letter also distinguishes this actual-phase obstruction fromphase-uniform obstructions and records supporting numerical diagnostics. The accompanying reproducibility package contains the exact trialcoefficients, rational integration intervals, Arb certificates,independent-recomputation verification code, execution logs, andsupplementary numerical programs. Rigorous calculations usepython-flint 0.9.0; floating-point diagnostics use NumPy 2.5.3and SciPy 1.18.1. These results provide a concrete mathematical motivation for combiningZero-Gram and positive-part methods while retaining the signedcontributions of the Weil quadratic form. The letter does not prove the Riemann hypothesis or establishpositivity of the full Weil form at support three.