Global Coercivity of Explicit–Formula Residuals A Rigidity Theorem for the Riemann Hypothesis

This paper presents a new operator-theoretic framework for analysing the Riemann Hypothesis through a coercivity–rigidity mechanism applied to the Guinand–Weil explicit formula. We construct a family of functional-equation–odd Gaussian probes and associate to them an explicit-formula residual realised as a Hilbert–Schmidt pairing between an antisymmetric defect operator and a probe operator. This residual vanishes identically for the true zero configuration of the Riemann zeta function but detects horizontal displacement of functional-equation pairs away from the critical line. The central analytic result is a global coercive inequality: the squared L2L^2L2 energy of the FE-odd residual dominates a weighted ℓ2\ell^2ℓ2 norm of the horizontal displacements of the zeros. The diagonal contribution is shown to be strictly positive, while the off-diagonal contribution is controlled using a square-sum estimate based only on unconditional inputs: the Riemann–von Mangoldt unit-scale counting law and a micro-scale anti-clustering bound derived from Jensen’s formula for the completed zeta function ξ(s)\xi(s)ξ(s). No spacing, repulsion, pair-correlation, or randomness assumptions are used. A probe-amplification argument employing disjoint frequency windows yields quantitative diagonal dominance without affecting admissibility or remainder bounds. The nonlinear remainder is shown to be uniformly controlled and arbitrarily small in the large-bandwidth limit. As a consequence, vanishing of the FE-odd residual forces rigidity of functional-equation pairs: all nontrivial zeros of ζ(s)\zeta(s)ζ(s) must lie on the critical line. The admissibility of the Gaussian probe family for the Guinand–Weil explicit formula is verified unconditionally for ζ(s)\zeta(s)ζ(s) and is independent of the location of its zeros. The argument implies the Riemann Hypothesis, frames it as a stability problem for the explicit formula and introduces a coercive energy method that may be adaptable to other LLL-functions satisfying analogous analytic properties.