What question did this study set out to answer?

The aim is to integrate the BRISM model with established quantum mechanics frameworks without changing the existing formalism.

February 14, 2026Open Access

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

Key Points

The aim is to integrate the BRISM model with established quantum mechanics frameworks without changing the existing formalism.
Developed four formal bridges embedding BRISM within Hilbert-space quantum mechanics.
Analyzed U(1) symmetry and its implications for norm conservation and the Born rule.
Examined Gleason–Busch measure uniqueness within POVM structures.
Applied Naimark–Stinespring dilation to relate every POVM to projective measurements.
Investigated spectral stability of quadratic density mappings under several conditions.
Confirmed that the Born rule is a necessary outcome of the BRISM interface rather than an assumption.
Proved that all POVM-induced probabilities align with standard probability rules.
Demonstrated the mathematical necessity of dilation spaces in relating bulk and measurement statistics.
Established that only quadratic mappings remain positive, local, and σ-additive across spectral components.

Abstract

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 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.

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

Key Points

Abstract

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