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

February 12, 2026Open Access

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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.

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Reveals how BRISM maintains the Born rule in quantum mechanics, clarifying measurable statistics.

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Swen Carlo Heinze (2026) studied this question.

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