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

TL;DR

This paper derives the quantum Born rule and formalism within Accessibility Theory by using boundary reduction and coherence assumptions, linking algebraic structures to quantum phenomena.

Contribution

It introduces a novel derivation of quantum formalism from boundary-based accessibility principles and algebraic constraints, connecting to the Standard Model and spacetime.

Findings

Derives the Born rule from boundary coherence conditions and Gleason's theorem.

Identifies the algebraic structure underlying the Standard Model and spacetime.

Shows how boundary reduction leads to quantum operational features.

Abstract

We show that the operational quantum formalism -- the Born rule, L\"uders 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.