An Unconditional Exponential Upper Bound for the Windowed Weil Quadratic Form

For an integer n ≥ 3 and half-window a = ½ ln n, let κ(n) be the infimum of the Rayleigh quotient Q(f)/‖f‖² of the windowed Weil quadratic form over all nonzero real-valued f in H₀¹(−a, a). We prove unconditionally that for every integer n ≥ 3, κ(n) ≤ C₃(1 + δ√n)·n^14.5·e^(−4πn), with C₃ = 3.03·10⁶ and δ ≤ 6.40·10⁻¹¹, and δ = 0 under the Riemann Hypothesis. An OEIS number (A401172) has been allocated for the integer sequence a(n) = ⌊−log₁₀ κ(n)⌋ (defined whenever κ(n) > 0). The proof is computer-assisted: it evaluates Q(f) on an explicit even witness f_n built, following Connes and Consani, as the Mellin–Poisson periodization of a three-term Kaiser–Bessel profile (ν ∈ {2, 3, 4}) with a constant boundary corrector, combining one integration by parts at the nontrivial zeros of ζ(s) with non-asymptotic Bessel bounds yielding the derivative tail bound ≤ 28·n⁵·e^(−2πn) for n ≥ 21 and the normalized norm bound ≥ 1.2·10⁻⁵ for n ≥ 1000 (with two analytical constants certified in ball arithmetic). The remaining finite range 3 ≤ n ≤ 1000 is closed by rigorous interval computations in Arb ball arithmetic, cross-checked by an independent implementation. A comparison table for n = 3…20 (18/18 cases) shows a margin of 4.28–12.01 orders of magnitude above certified upper bounds U(n) ≥ κ(n), obtained by a different method. Only an upper bound is proved; we do not address lower bounds (κ(n) ≥ 0 for all n) and make no claim toward the Riemann Hypothesis.