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

Abstract

We present a spectral framework for gauge structure and fermion content. The basic datum is a discrete spectrum with finite multiplicities, generating a mass tower. Physical quantities are defined by convergent spectral sums and closed analytic expressions. The construction starts from a disorder-dominated coarse-graining principle, a minimal unbiased spin-1/2 change. This compact specification encodes the symmetry group, the chiral carrier, and the Clifford module. Its chiral structure selects RP3 = S3/Z2 within the stated minimal quotient criterion. The associated Clifford module has dimension four, and the scale flow gives a 3+1 decomposition. Condensation in the right sector reduces the isometry algebra so(4) isomorphic to su(2)L ⊕ su(2)R to su(2)L ⊕ u(1)Y. An additional colour carrier, admitted as structural datum, is fixed by minimality to su(3). The construction is conditional: the coarse-graining principle fixes the primitive chiral carrier, while the colour carrier, matter completion, and Higgs sector are stated as structural closures rather than absorbed into the principle. The gauge field contains a colour-singlet sector. The formulation specifies conditions for the spectral datum, physical quantities, state space, and mapping to field-theoretic structures. The free sector is recovered via Gaussian Osterwalder-Schrader reconstruction. The interacting sector is described by the spectral correspondence and spectral sums. Each ingredient is classified as theorem, definition, convention, or assumption.Keywords: Spectral datum; Scale flow; Gauge structure; Positive scalar mass; Coarse-graining principle; Encoding; Osterwalder-Schrader reconstruction; Spectral sums