Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau-Widom decay law

Abstract

Weil's criterion states that the Riemann Hypothesis (RH) is equivalent to the non-negativity of an explicit quadratic form Q(f) on test functions. For f supported in [-L,L] we study the normalized infimum lambda_min(L) = inf Q(f)/||f||^2 from both sides. Lower bounds (unconditional): positivity on bounded support is classical for supp f in [-(log 2)/2,(log 2)/2] (Yoshida; Connes-Consani). We prove by certified computation that Q(f) >= 8.9e-18 ||f||^2 for all supp f in [-0.8,0.8], i.e. autocorrelation support 1.6, 2.3 times the classical range. The proof rests on a one-stroke reduction: a pointwise envelope for the Weil symbol, with a comb constant shown optimal by Weyl equidistribution, converts positivity on the whole window into positive semidefiniteness of a single finite matrix. The odd parity sector shows that the certified positivity holds for arbitrary complex test functions of support 1.6, and that the window ground state is simple and even, the spectral hypothesis required by the operator-theoretic program of Connes, Consani, Moscovici and van Suijlekom. Upper bounds (unconditional): variational bounds are evaluated on the geometric side in interval arithmetic, with no appeal to zeros or RH, down to 3.2e-283 at L=2. They follow the empirical law -ln lambda_min(L) ~ 2 pi^2 N(T*)/ln N(T*), T* = 2 pi e^{2L}, with N the zero-counting function; 2 pi^2 matches the Landau-Widom eigenvalue-plunge rate. Under RH, lambda_min(L) <= exp(-L e^L) for all large L. Synthesis: at L=0.8 the two halves enclose the profile, 8.9e-18 <= lambda_min(0.8) <= 2.27e-17, both certified. We show why the positivity route cannot reach RH unassisted: any one-stroke certificate must resolve frequencies up to 2 pi e^{A_L}, A_L ~ 4e^L, a doubly exponential threshold that no pointwise bound on the prime comb lowers, while the spectral margin collapses at the Landau-Widom rate.