Standard-Model Interactions over Finite Relational Substrate
Abstract
The electromagnetic, weak, and strong interactions, together with one Standard-Model generation of matter, arise on a finite relational arithmetic substrate \(\mathbb F_\Omega\) as the gauging of its relational frame data: a cell-local change of frame is a gauge symmetry and the compensating connection is the field. The three interactions are the substrate's three internal frame data: the multiplicative phase, the spinorial extension, and the colour frame. A finite-window correspondence ties the finite gauge groups so produced to the observable compact theory: a cyclotomic representation reproduces the gauge invariants, the Yang-Mills kinetic operator, and the chiral matter within a bounded horizon \(H\ll\Omega\) at residue \(O(H/\Omega)\), proven on the resolvable class, the cross-scale extension a finite Carrier-scale residue. Concretely, electric charge is a quantised winding index and the bare coupling the phase-channel capacity \(1/4\pi\); electroweak breaking gives the massless photon, custodial \(\rho=1\), and sin2 θW=3/8; colour SU(3) is the special unitary group of a Hermitian three-form; and one generation is the anomaly-free SO(10) spinor 16 with a right-handed neutrino. The four interactions and three generations are organised by one closed role ladder (the count/generate split), forbidding a fifth force and a fourth generation and forcing SO(10) unification. Every exact claim is verified in finite-field or cyclotomic arithmetic, the continuum entering only as a…
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