A Coercive Quadratic Form from the Riemann–Weil Explicit Formula

This paper introduces a coercive quadratic form canonically induced by the Riemann–Weil explicit formula. The form acts on a natural Sobolev test space and decomposes into an archimedean contribution together with a remainder term encoding prime and zero effects. The main result establishes a strong coercivity property: the existence of any single nontrivial zero of the Riemann zeta function off the critical line forces the value of the quadratic form to diverge. This divergence arises from an explicit exponential mode associated to the real part of the zero and cannot be cancelled by interference, averaging, or collective effects. No assumptions are made about the location of zeta zeros. The analysis relies only on unconditional ingredients, including the Riemann–Weil explicit formula, elliptic coercivity, classical Fourier estimates, and unconditional zero-density bounds. The result isolates a structural obstruction: off-critical zeros are incompatible with finite admissible energy. This coercive mechanism serves as the analytic foundation for subsequent developments. In particular, it is lifted to a uniform operator-theoretic setting in Uniform Coercivity for the Báez–Duarte Frame Operator (Zenodo DOI: 10.5281/zenodo.18529293), and later combined with canonical-system rigidity arguments in Energy Rigidity and Canonical Forcing of the Riemann Zeta Zeros (Zenodo DOI: 10.5281/zenodo.18529295) and Coercivity, Canonical Systems, and Spectral Rigidity (Zenodo DOI: 10.5281/zenodo.18529297).