This paper develops four formal bridges that embed the BRISM interface model within the established Hilbert‑space framework of quantum mechanics. BRISM describes how real, normalized measurement statistics on the brane arise from complex bulk amplitudes through a phase‑neutral interface mapping, without modifying the standard formalism or introducing additional ontology. (i) U(1) symmetry & Noether: The global phase invariance of Schrödinger dynamics enforces norm conservation and identifies the quadratic Born rule as the unique phase‑invariant, norm‑preserving mapping from amplitudes to measurable densities.(ii) Gleason–Busch measure uniqueness: All POVM‑induced probabilities arising through the interface naturally conform to the noncontextual measure structure required by the standard probability rules of quantum mechanics.(iii) Naimark–Stinespring dilation: The bulk corresponds to the mathematically necessary dilation space in which every POVM becomes a projective measurement; it is not an added physical assumption but the structural completion of the measurement formalism.(iv) Spectral stability (new): Only quadratic density mappings remain compatible with positivity, locality, σ‑additivity, and phase neutrality across spectral components, making the Born rule a structural necessity of the interface rather than a postulate. Overall, the work reorganizes the standard framework internally, clarifying how observable statistics emerge from the structural properties of the bulk–brane interface. It extends the conceptual basis introduced in the author’s earlier BRISM papers (the foundational BRISM paper DOI: 10.5281/zenodo.18391944, and further summarized in the concise BRISM overview DOI: 10.5281/zenodo.18491724).