Toward the Origin of Quantum Probability
Year: 2026
Type: preprint
Abstract: Toward the Origin of Quantum Probability investigates the physical foundations of quantum probability within Pattern Field Theory (PFT), establishing an exact conservation and local Transport-current law while examining how the continuous distribution of quantum amplitudes relates to discrete, exclusive measurement outcomes. Building upon Quantum Gravity Fully Derived, the phase-valued Allen Orbital Lattice (AOL), and the relational quantum-field architecture of PFT, this paper develops a mathematical and computational account of phase-dependent Transport, conserved quadratic participation, detector interaction and the conditions required for physical measurement commitment. The central result is an exact local continuity equation derived from Hermitian lattice Transport. The construction establishes conservation of quadratic density, identifies the directed currents responsible for transferring participation between relational configurations, and demonstrates how quantum interference is preserved within a closed physical system. Principal contributions include: Exact Transport Conservation: Derives the local continuity equation for the phase-valued AOL operator and establishes global conservation of quadratic density under Hermitian Transport dynamics. Origin of the Quadratic Measure: Extends the inherited Quadratic Measure Lemma, examining the mathematical conditions under which a conserved quadratic measure arises and its relationship to the probability structure of quantum mechanics. Phase-Dependent Transport Currents: Establishes the relationship between relational phase, directed boundary currents and the redistribution of conserved participation across detector regions. Closed-System Detector Calculation: Presents a reproducible numerical investigation of a closed Hermitian Transport configuration containing a phase-bearing ring coupled to three detector channels. Conservation at Numerical Precision: Demonstrates quadratic norm conservation with maximum numerical deviations of approximately 1.8 x 10^-14 across the examined configurations. Quantum Interference: Reproduces phase-dependent detector distributions, including destructive interference at the central detector, where the calculated participation approaches 3.8 x 10^-30. Detector Retention and Return Currents: Identifies transient reverse currents and demonstrates why participation transferred into a detector must be distinguished from a permanently retained measurement record. Cut-Invariance: Demonstrates that the asymptotic detector participation pattern becomes independent of the chosen system-detector partition in the completed-transfer limit, while transient measurements retain partition dependence. Transport Versus Physical Commitment: Establishes the distinction between conserved, continuously distributed quantum participation and the exclusive physical commitment required for an individual detector outcome. Native EQUI Allocation: Identifies the physical connection required between phase-sensitive Equilibrion (EQUI) participation, inherited detector configurations and the eventual frequency distribution of committed events. Conditional Outcome Correspondence: Examines deterministic successor selection using phase-sensitive admissible currents and inherited route residuals, identifying the equilibrium conditions under which the resulting distributions reproduce quadratic outcome weighting. Reproducible Computational Evidence: Includes the Python source code and machine-readable numerical results used in the closed-Transport detector investigation, permitting independent examination of conservation, current integration, interference and detector participation. The paper identifies a precise remaining physical question: how does native Transport convert distributed, conserved participation into one retained measurement outcome per preparation, with observed frequencies determined by the appropriate quadratic measure? The mathematical conservation law, phase-dependent currents, detector calculations and asymptotic participation results are established within the specified Hermitian construction. The physical identification between native EQUI participation and spectral density, together with the commitment rule governing exclusive records, remains the explicitly defined closure problem. A Derivation Ledger distinguishes exact mathematical results, inherited PFT constructions, conditional correspondence results and the physical commitments requiring further derivation. The paper also examines operational reconstructions of quantum theory, composite systems, correlations, no-signalling conditions and the relationship between native causal continuation and downstream Mensural spacetime. The significance of this work lies in connecting quantum probability to a physically specified conservation and Transport architecture while isolating the exact mechanism still required to explain individual measurement outcomes. It establishes a reproducible mathematical and computational foundation for investigating the origin of the Born measure, detector commitment and the transition from distributed quantum participation to observed physical events. Together with Recovering General Relativity in Pattern Field Theory and Recovering Quantum Field Structure in Pattern Field Theory, this work extends the common relational Transport architecture into quantum conservation, probability and measurement. James Johan Sebastian AllenPattern Field Theoryhttps://patternfieldtheory.com/https://zenodo.org/communities/pft
Author: Allen, James Johan Sebastian
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225315401