Published February 12, 2026 | Version 1.45 E
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B03: BRISM and the Born Rule in Continuity with Established Structural Theorems of Quantum Mechanics: U(1) Symmetry, Measure Uniqueness, POVM Dilation and Spectral Stability

Authors/Creators

Description

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.

Status note (May 2026):

This paper predates BRISM Status 3M and should be read within the current BRISM claim hierarchy. In Status 3M, the Born readout density and the minimal phase layer U(1) remain part of the hard current core of BRISM. See: BRISM Status 3M: Logical Consolidation and Claim Hierarchy

For a current top-down overview of the BRISM framework and its wider structural context, see: Current BRISM Top‑Down Onepager

All BRISM papers on Zenodo  >> Searchlist
 
Supplementary structural material, dependency maps, and reproducibility notes are available in an accompanying public repository:
https://github.com/swencarloheinze/brism-framework

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Dates

Submitted
2026-02-12
Bridge V 1.45

References

  • A. M. Gleason, Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Mechanics, 6 (1957). P. Busch, Quantum states and generalized observables: a simple proof of Gleason's theorem, Physical Review Letters, 91, 120403 (2003). M. A. Naimark, Spectral functions of a symmetric operator, Izvestiya Akademii Nauk SSSR, Seriya Matematicheskaya, 4 (1940). W. F. Stinespring, Positive functions on C*‑algebras, Proceedings of the American Mathematical Society, 6 (1955). W. Pauli, Exclusion principle, Lorentz group and reflection of electrons, in: Niels Bohr and the Development of Physics, Pergamon Press (1955). E. Noether, Invariante Variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen (1918). P. Jordan and J. von Neumann, On inner products in linear, metric spaces, Annals of Mathematics, 36(3):719–723 (1935). A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, 2nd Edition, Edizioni della Normale (2011).