Quantum Foundations and Particle Physics in the Universal Model Framework From Distinction Logic to Gauge Symmetries: Prime-Indexed Information Geometry as the Root of Physical Law

This paper provides a comprehensive formulation of quantum mechanics and the Standard Model of particle physics within the Universal Model Framework (UMF), where spacetime and matter emerge from a prime-indexed fractal information geometry. We construct prime-indexed Hilbert spaces Hπ\mathcal{H}_\piHπ and show that the discrete prime Laplacian yields the Schrödinger equation via a lattice action principle. Measurement theory follows from orthomodular prime-projector lattices, reproducing the Born rule through a Gleason-type theorem adapted to the prime substrate.

We derive enhanced entanglement scaling, Sπ(L)∼ln⁡(πL)ln⁡LS_\pi(L)\sim\sqrt{\ln(\pi L)}\ln LSπ(L)∼ln(πL)lnL, reflecting multipartite connectivity on the prime lattice. A Lindblad treatment of stochastic phase diffusion predicts decoherence behavior in agreement with trapped-ion and NV-center experiments (within 5–10%). Prime residues modulo 6 give rise to the Standard Model gauge group, Yukawa couplings, Higgs vacuum structure, and anomaly cancellation conditions. LHC Run-2/3 data leave several predicted prime-harmonic scalar resonances (54, 200, 690 GeV) unexcluded. The UMF reproduces 16 fundamental constants with >85% exact or excellent agreement. This work positions prime arithmetic as a viable generative substrate for quantum theory, gauge symmetry, and particle mass spectra.

• This project was developed by Marco Gericke, with structured assistance from a large language model. All scientific concepts and conclusions were generated, verified, and interpreted by the author.
• Dedicated to Peter Plichta, who envisioned the code before it could be computed.