Derivation of the Born Rule and Operational Quantum Formalism in the Accessibility Framework through Boundary Reduction

• URL: http://arxiv.org/abs/2604.27125v2
• Date: Fri, 01 May 2026 03:00:46 GMT
• Title: Derivation of the Born Rule and Operational Quantum Formalism in the Accessibility Framework through Boundary Reduction
• Abstract summary: We show that the operational quantum formalism -- the Born rule, Lders state updating, quantum interference, non-Markovian effective dynamics, and Bell inequality violation at the Tsirelson bound $2sqrt2$ -- arises within Accessibility Theory (AT)<n>AT is a framework built on real graded spectral triples and a single algebraic selection principle.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show that the operational quantum formalism -- the Born rule, Lüders state updating, quantum interference, non-Markovian effective dynamics, and Bell inequality violation at the Tsirelson bound $2\sqrt{2}$ -- arises within Accessibility Theory (AT) from the Aperture construction together with explicit coherence and locality assumptions stated in the paper. AT is a framework built on real graded spectral triples and a single algebraic selection principle. The Principle of Universal Accessibility Balance requires three independent measures of the complexity of a spectral triple -- its algebraic, gauge-theoretic, and geometric content -- to be exactly equal and minimized, uniquely selecting the algebra $\mathbb{C} \oplus \mathbb{H} \oplus M_3(\mathbb{C})$ and with it the Standard Model gauge group, particle content, four-dimensional Lorentzian spacetime, three generations, and gravitational dynamics. Restriction to a codimension-one geometric boundary reduces this algebra to its commutative center $\mathbb{C} \oplus \mathbb{C} \oplus \mathbb{C}$ -- the Aperture -- which defines a permanent information bottleneck for any embedded observer. Coherence conditions on inference through this bottleneck, together with Gleason's theorem on the 48-dimensional internal Hilbert space, uniquely determine the Born rule; the remaining operational features follow from the same observer-level framework under the stated assumptions. At the ontological level the theory is deterministic and state-realist, while the operational quantum formalism appears at the observer level as a consequence of structurally limited access to the underlying algebraic reality.

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