Standard-Model Interactions over Finite Substrate

We show that the electromagnetic, weak, and strong interactions, together with one complete generation of Standard-Model matter, arise on a finite arithmetic substrate as the gauging of its relational frame data, where a cell-local change of frame is a gauge symmetry and the connection that compensates it across the substrate is the field. The three interactions are the substrate's three internal frame data — the multiplicative phase, the spinorial extension, and the colour frame — each gauged by this single argument. Electromagnetism is built exactly: electric charge is a quantised winding index, the photon is massless because the drive conserves the phase cycle, the Coulomb law is gravity's own lattice Green's function differing only in sign and spin. Furthermore, Maxwell is the unique relevant gauge action, where gauge invariance itself annihilates the lattice's anisotropy. The bare coupling is the phase-channel capacity \(1/4\pi\), and the \(10^{36}\) electromagnetic-to-gravitational hierarchy is a reading of the substrate size \(\Omega\sim10^{122}\). Electroweak breaking is the misalignment of the split torus carrying the drive and the non-split torus carrying isospin, giving the massless photon, the custodial \(\rho=1\) exactly, and the Weinberg angle \(\sin^2\theta_W=\mathrm{Tr}(T_3^2)/\mathrm{Tr}(Q^2)=3/8\) for a complete generation. Colour \(\mathrm{SU}(3)\) is realised as the special unitary group of a Hermitian three-form over the spinor extension, confinement its compact-group area law, with the \(\mathbb{Z}_3\) triality centre present when \(\Omega\equiv2\pmod3\). One complete generation is derived as the spinor \(\mathbf{16}\) of the rank-five internal frame \(\mathbb{C}^3\oplus\mathbb{C}^2\), including all its gauge representations, hypercharges, charges, anomaly freedom, and a right-handed neutrino. Finally, the four interactions are the four primitive arithmetic roles: their closure forbids a fifth force, and its generative/non-generative split yields one bosonic carrier and exactly three fermion generations, with simple unification \(\mathrm{SO}(10)\) forced by the same relational ontology and chirality fixed by the drive. Every exact claim is verified in finite, finite-field, or cyclotomic arithmetic; the continuum enters only as a labelled degenerate idealisation.