The spectrum of a compact internal space I: Gauge structure and fermion content

We study how internal geometry, gauge response and matter representations can arise from minimal unbiased change, and how their spectral response can support condensation and a mass gap. The construction treats spacetime localization as emergent and starts with a real orthogonal carrier of change. Independent composition and quadratic scaling determine a Gaussian semigroup. Its correlation profile fixes a finite reflecting window, while a separate quadratic closure gives the ladder half-step and central sign. Under stated physical response principles, these data yield a three-dimensional projective response space, colour, weak-isospin and phase symmetries with their common finite quotient, and a sixteen-dimensional matter representation with a fifteen-dimensional charged quotient. The elementary connector is a single Higgs-type doublet. Mode multiplication and matrix commutators supply non-Abelian coupling and matter transport. For the specified spectral feedback and reference normalization, we prove convergence and stability of the complete negative-logarithmic potential and uniqueness of its positive amplitude square in the condensed regime. Kinetic normalization and the framework’s scalar mass assignment then give a sharp positive gap for the full scalar tower, including its constant mode. Uniform spectral tail bounds make this result accessible to finite truncation and scale readout. The gap has a colour-singlet Yang–Mills/glueball interpretation; field-theoretic reconstruction and algebraic consistency checks provide complementary realizations of the generated structure.