* * * This version has been superseded. Please use version 1.28 E instead. * * * * * * link: https://doi.org/10.5281/zenodo.18607697This technical supplement extends the BRISM framework introduced in BRISM v1.43, demonstrating that the three gauge groups of the Standard Model emerge naturally from structural stability conditions at the bulk–brane interface. Building on the established derivation of the Born rule from phase neutrality, positivity, and spectral stability, we show: Phase stability of the interface enforces a global SU(1) ≃ U(1) symmetry. Rotational continuity of the projection map requires the universal covering group SU(2), reproducing spin‑½ structure as a topological necessity. Dimensional spectral stability of a three‑dimensional complex brane embedded in an infinite‑dimensional dilation space selects SU(3) as the minimal stabilizing group. Together, these results provide a structural origin for 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 the internal consistency of the BRISM interface. The analysis suggests that gauge symmetries arise as geometric and topological stabilizers of the measurement interface rather than as fundamental input, offering a unified perspective on the emergence of quantum‑field‑theoretic structure from Hilbert‑space geometry.