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

We prove a global coercivity theorem for FE–odd explicit-formula residuals associated with zeta-compatible zero configurations. The coercive inequality forces rigidity of the horizontal displacements of functional-equation pairs, implying that all nontrivial zeros lie on the critical line and thereby establishing the Riemann Hypothesis. The argument is purely analytic and relies on no spacing, repulsion, or pair-correlation assumptions. Off-diagonal interactions are controlled via weighted Schur estimates using only the unconditional Riemann–von Mangoldt local zero-counting bound, while a quantitative probe-amplification mechanism enforces strict diagonal dominance. The nonlinear remainder is uniformly controlled and becomes negligible at large probe bandwidth. The admissibility of the FE-odd Gaussian probe family for the Guinand–Weil explicit formula is verified unconditionally and independently of zero location. No Hilbert–Schmidt convergence or operator compactness assumptions are required. Earlier work by the author developed related explicit-formula frameworks and stability heuristics; the present paper consolidates and completes these ideas into a closed coercivity–rigidity argument yielding the Riemann Hypothesis.