The Canonical Spectral-Curvature Field: A Unified Operator Origin for Coherence, Dissipation, and Spectral Budgets

This paper develops the Canonical Spectral-Curvature Field (CSCF) as a unified operator framework within Shadow Theory’s physical source-to-readout architecture. One positive selfadjoint spectral carrier supports complementary coherent and dissipative responses: reversible phase evolution and irreversible attenuation share the same underlying spectral structure. The construction begins with a closed nonnegative curvature form and a specified Hilbert-space metric. From these data, the paper establishes the common carrier, its functional-calculus responses, exact recovery from averaged dissipative transmission, spectral budgets, and conditions for observable reconstruction and dynamical closure. It unifies the earlier Lambda-field and coherence-field constructions in one mathematical treatment. A central contribution connects this architecture to the recurring spectral-ratio structures documented in the companion cross-domain investigation of hydrogen coupling graphs, gravitational Kerr spectra and image-derived graph Laplacians. The ordered ratios are identified exactly with the spectrum of a finite-dimensional modular operator. Their logarithmic differences generate a reversible conjugation flow, giving the measured profiles an explicit operator relationship to the coherent face while retaining their common spectral carrier. A quantitative closure-to-coherence theorem bounds coherent leakage using a dissipative closure deficit under stated bandwidth and regularity assumptions. Worked realizations include mobility-weighted curvature, reciprocal storage and backreaction, hidden frequency-dependent response, detailed-balance dynamics, aligned complex-gradient evolution and boundary-induced operators. The paper preserves the cross-domain study’s positive recurrence and robustness findings alongside its domain differences, readout dependence and library-specificity results. Complete derivations and computational verification connect these findings to CSCF’s role between admitted source realization and observable physical readouts. Deriving the particular recurring profiles from physical realization laws is identified as the next stage of the programme.