Energy Rigidity and Canonical Forcing of the Riemann Zeta Zeros

This paper proves that uniform coercivity of the relevant operator forces spectral confinement through canonical-system rigidity. Assuming the uniform coercivity result established in Uniform Coercivity for the Báez–Duarte Frame Operator (Zenodo DOI: 10.5281/zenodo.18529293), we show that no non-real spectral points are admissible in the associated canonical system. The argument proceeds by passing from coercivity of the frame operator to resolvent positivity, yielding a Herglotz function whose representing measure has support bounded away from zero. Using standard inverse spectral theory for canonical systems, this support condition forces the Hamiltonian to be coercive almost everywhere. Coercive Hamiltonians are limit-point at infinity, and therefore admit no non-real spectrum. The Riemann ξ-function enters only at the terminal identification stage, as boundary spectral data within the de Branges classification of canonical systems. Individual zero locations are never assumed or used as inputs. This paper establishes the rigidity implication: once uniform coercivity is granted, off-critical zeros are forbidden by operator-theoretic necessity. The result prepares the ground for the final synthesis and closure carried out in Coercivity, Canonical Systems, and Spectral Rigidity (Zenodo DOI: 10.5281/zenodo.18529297).