Two Complementary Methods for Weil Positivity at a=3

This paper investigates two complementary approaches to Weil positivityat Fourier support a = 3: the Zero-Gram method, based on verified zerosof the Riemann zeta function, and the exact-phase Positive-Part method,based on the arithmetic structure of the Weil quadratic form. On the Zero-Gram side, we develop explicit lower bounds for finite-dimensionalGram matrices using selected critical-line zeros, Cauchy matrix estimates,and computer-assisted certification. Subject to stated external inputsand interval-arithmetic certificates, we obtain the exponential bound \[\lambda_{\min}(W_M^{\rm zero}) \ge e^{-11.86n}\] over an explicit dimension range extending to approximately3.896 × 10^11. We also establish an unconditional two-sided truncation estimaterelating the finite zero Gram matrix to the Weil quadratic form,and certify positivity on several low-dimensional sine-blocktrial spaces without assuming the Riemann hypothesis. On the Positive-Part side, we incorporate a phase-uniform high-frequencyoperator estimate above 3.26 × 10^22, while carefully distinguishingthis result from positivity of the complete Weil form. The principal methodological contribution is an exact localizationframework based on a bandlimited tight partition. This yields anabsolutely convergent decomposition of the Weil quadratic form intolocal contributions, without introducing approximation errors betweenthe two representations. Within this framework, we formulate hybrid certificates combininglocal Zero-Gram estimates, Positive-Part bounds on complementarysubspaces, and Schur-complement inequalities controlling cross terms. The resulting analysis separates the remaining difficulties intotwo explicitly formulated problems: 1. The Detection-Horizon Problem: whether every failure of the Riemann hypothesis must produce a negative Weil direction detectable at support no greater than three. 2. Buffered Complementary Coverage: whether all localized contributions can be certified nonnegative using the complementary methods at a slightly enlarged support. We prove that resolving both problems would imply the Riemann hypothesis. A Nyquist-type analysis further clarifies the limitations of directhigh-zero detection and identifies possible collective, nonlocal,and arithmetic mechanisms relevant to the Detection-Horizon Problem. The significance of this work lies in combining quantitativecomputer-assisted estimates with an exact operator-theoreticlocalization framework, and in distinguishing established resultsfrom the specific analytic and constructive conditions still neededfor a complete proof. This paper does not prove the Riemann hypothesis. The Detection-Horizonstatement and the constructive coverage of all local cells remain open.