Spectral Harmonics as Resonance Eigenvalues Harmonic Closure, Coherence Selection and Curvature Emergence

This paper develops a closure-based account of spectral harmonics, resonance eigenvalues, andrealized harmonic structure. The central claim is that spectral harmonics should not be identified simply with eigenvalues, but with the manifested forms of resonance eigenmodes indexed by eigenvalues and stabilized under closure conditions. A resonance eigenvalue gives spectral permission; an eigenmode gives structured possibility; a spectral harmonic is the realized appearance of that possibility under coherence selection and closure stabilization.To formalize this distinction, the paper introduces a minimal operator architecture consisting ofa resonance operator R, a coherence selection operator C, a closure functional F, and aharmonic manifestation map H. A mode becomes realized only when it satisfies spectraladmissibility, coherence selection, and closure sufficiency. This yields the central realizationcondition: R psi = lambda psi, C(psi) != 0, and F[psi] >= k_cl. The paper further defines a composite realization operator Theta_cl = H o C o P_cl andproposes a Harmonic Closure Index as a first scalar measure of realization tendency. These toolsallow spectral possibility to be distinguished from harmonic persistence. The framework is thenextended across atomic shells, gauge-sector differentiation, and curvature emergence. In thegeometric domain, curvature is interpreted as the projected trace of closure-stabilized harmonicrealization rather than as a primitive substrate. Keywords spectral harmonics; resonance eigenvalues; harmonic closure; coherence selection; closurefunctional; spectral closure law; resonance operator; curvature emergence; gauge structure;atomic ontology; closure mathematics