Eigenmodes of the Informational Curvature Functional: A Structural Origin for the Spectrum of Matter

This paper develops the eigenmode structure of the informational curvature functionalKI=F F^ derived from the induced‑metric constraintg = F C. Linearizing around a stable configuration yields a second‑variation operator whose eigenfunctions represent the natural excitations of the informational field. We show that this operator possesses a discrete spectrum and propose that these eigenmodes correspond to the observed spectrum of matter. This interpretation unifies weak‑field galactic relaxation, strong‑field black‑hole ringdown, and particle‑level structure under the same five informational‑geometric equations. The resulting picture suggests that matter arises not from additional fields or symmetries, but from the intrinsic eigenstructure of the informational curvature functional. This work forms a central component of the Universal Relaxation Experiment and the broader informational‑geometric framework. Math StatementAll mathematical expressions in this manuscript are presented in duplicate form. The bold LaTeX expressions constitute the canonical, archival mathematical text. Each bold LaTeX expression is preceded by a Unicode inline translation provided solely for compatibility across viewing environments. In cases where minor typographical differences arise between the two renderings, the bold LaTeX expression takes precedence as the authoritative mathematical content. Conical Source StatementThis paper is one facet of a larger informational‑geometric architecture. Each manuscript in the series represents a distinct cross‑section of the same underlying conical source: the five structural equations that govern weak‑field relaxation, strong‑field ringdown, cosmological deceleration, and the eigenmode structure of matter. The present work should therefore be understood as part of a unified framework rather than an isolated hypothesis.