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.

Quantum Foundations and Particle Physics in the Universal Model Framework (UMF) develops a single, axiomatic route from distinction logic to quantum mechanics, the Standard Model gauge group, and a one-parameter gravitational sector. The framework models physical reality as a prime-indexed information lattice M_π, from which a prime Hilbert space H_π and orthomodular lattice L(H_π) are constructed; the Schrödinger equation then appears as the geodesic equation on M_π (Theorem 1), and the Born rule follows from Gleason’s theorem (Theorem 2). Standard Model gauge symmetry [SU(3)×SU(2)×U(1)]/Z₆ is derived from residue classes mod 6, twin primes, and prime triplets (Theorem 3), with U(1) anomalies canceled via a Green–Schwarz axion whose decay constant lies in the conventional invisible window and is fixed by anomaly matching. A rank‑1 Jacobian in the gravitational sector collapses four apparent parameters to a single effective coupling λ_eff, from which α⁻¹, G_eff, Λ, and Lorentz-violation bounds |u*–1| are jointly reproduced with high precision. The paper further unifies Gödel incompleteness, operational calibration, and measurement contextuality into a “Structured Incompleteness as Operational Necessity” principle, and applies the prime‑lattice geometry to the proton-radius anomaly, reproducing the e–μ discrepancy at the 0.1% level with zero free parameters and predicting a distinct τ‑hydrogen radius as a sharp falsifiable test. 

New Section 8: Electroweak Symmetry Breaking from Twin-Prime Condensation.

• 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.