BRISM and the Born Rule in Continuity with Established Structural Theorems of Quantum Mechanics: U(1) Symmetry, Measure Uniqueness, POVM Dilation and Spectral Stability

This paper develops four formal bridges that embed the BRISM interface model into the established Hilbert‑space structure of quantum mechanics. BRISM describes how real, normalized measurement statistics on the brane arise from complex bulk amplitudes through a phase‑neutral interface.

(i) U(1) symmetry and Noether’s theorem identify the quadratic Born rule as the unique phase‑invariant, norm‑preserving mapping from amplitudes to measurement densities.
(ii) Gleason–Busch measure uniqueness shows that all POVM‑induced probabilities constructed through the interface naturally conform to the noncontextual measure structure of quantum theory.
(iii) Naimark–Stinespring dilation establishes the bulk not as an added physical assumption but as the mathematically required extension of POVMs to projective measurements on a larger Hilbert space.
(iv) Spectral stability (new) demonstrates that only quadratic density mappings remain compatible with positivity, locality, σ‑additivity, and phase neutrality across spectral components, selecting the Born rule as a structural necessity of the interface rather than a postulate.

Overall, the work reorganizes the standard quantum‑mechanical framework without introducing new ontology, clarifying how measurement statistics arise from the structural properties of the bulk–brane interface. It extends the conceptual foundations 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).