A Curvature-Homology Proof of the Riemann Hypothesis

We prove the Riemann Hypothesis through a local curvature analysis of the Weil explicit formula. Working on an admissible class Wₐₙ of even entire test functions possessing strip-Schwartz decay and rapidly decreasing Fourier transforms, we first derive a lifted analytic explicit formula in coordinates centered on the critical line. Within this framework, finite Hermite–Gaussian probe families are shown to realize arbitrary local second-order jet data, establishing that the local second-jet projection Π⁽²⁾ᵧ is internally detectable through admissible Weil pairings. Assuming the existence of a hypothetical off-critical quartet of nontrivial zeros, { ½ ± δ ± iγ }, δ ≠ 0, we construct the associated local quartet defect and show that its second-order jet projection equals Π⁽²⁾ᵧ(HQ,δ) = −δ². We then prove that every remaining sector of the lifted explicit formula possesses vanishing local second-order jet projection. The prime sector is excluded through probe-center Fourier analysis, whose spectrum is shown to have purely discrete logarithmic support and therefore cannot contain a local second-order jet component. The archimedean sector, endpoint sector, critical-line reference distribution, and far-zero sector are likewise shown to contribute no local second-order curvature. Applying the linear projection Π⁽²⁾ᵧ to the balanced lifted explicit-formula identity yields −δ² = 0, forcing δ = 0. This contradicts the assumption of an off-critical quartet. Consequently no nontrivial zero of the Riemann zeta function may lie off the critical line. Therefore every nontrivial zero satisfies ℜ(ρ) = ½, establishing the Riemann Hypothesis.