Technical Supplement: Interface Stability and the Induction of U(1), SU(2), and SU(3) Symmetry Structures in the BRISM Framework

This technical supplement extends the BRISM framework introduced in BRISM v1. 45 by demonstrating that the three gauge groups of the Standard Model arise from structural stability requirements at the bulk–brane interface. Building on the established derivation of the Born rule from phase neutrality, positivity, and spectral stability, the present analysis identifies three successive stability layers: Phase stability of the interface enforces invariance under global complex phases, yielding the internal symmetry group U (1). Rotational interface stability requires that spatial rotations lift consistently through the interface, selecting the universal covering group SU (2) and reproducing the spin‑½ structure as a topological consequence of the double cover of SO (3). Dimensional spectral stability of a three‑dimensional complex brane embedded in an infinite‑dimensional dilation space motivates SU (3) as the minimal group preserving complex volume, unitarity, and the spectral geometry of the interface. Taken together, these stability layers form a topological attractor whose fixed point is the Standard Model gauge symmetry SU (3) c × SU (2) L × U (1) Y, showing that these groups need not be postulated independently but follow from increasingly strong stability constraints inherent to the BRISM interface. This perspective suggests that gauge symmetries emerge as geometric and topological stabilizers of the measurement interface, offering a unified Hilbert‑space–based explanation for core features of quantum field theory.