UCM 49 — Emergent SU(3) × SU(2) × U(1) from the UCM Substrate: Bulk Modes, Membrane Oscillations, and Lattice Geometry

The Standard Model gauge group SU(3) × SU(2) × U(1) emerges from the UCM substrate without postulating gauge invariance. Three mechanisms contribute: (i) the quartic self-interaction of the transverse bulk field, constrained by div(psi)=0, is exactly U(1)-invariant (proved), with breaking only at RG-irrelevant order 6; (ii) each cell membrane carries two tangential oscillation modes whose commutation relations generate the su(2) Lie algebra via the Schwinger construction (proved), with SU(2) not SO(3) from half-integer Fock space representations; (iii) the three membrane orientations of the cubic lattice provide a colour triplet whose democratic edge-coupling is SU(3)-invariant at quartic order (conjectured). Local gauge invariance follows automatically from the Wilson plaquette structure of the link variables (proved). The coupling hierarchy kappa_mem >> kappa_cell gives g_3 > g_2 > g_1, matching the Standard Model. Chirality is argued to arise from helicity-selective coupling to circular membrane polarisations. The status of each result — proved, plausible, or open — is stated explicitly.