Revised Claim Boundary for the H4/GZ60 Discrete Algebraic Framework

This work presents a calibrated description of the H₄/GZ60 discrete algebraic framework, a structural model that connects a 60-state cyclic system to the geometry of the H₄ 600-cell and to algebraic structures appearing in particle physics and atomic spectra. The framework is based on the factorization60 = 12 × 5interpreted as a ℤ₁₂ × ℤ₅ cyclic structure, which organizes the state space into twelve ring types and five hierarchical layers. Within this structure the paper separates results into three levels of claim strength: • Theorem-level result: An exact discrete SO (4) Casimir derivation reproducing the hydrogenic energy spectrumEₙ = −1/ (2n²). • Algebraic-audit results: Exact generator-level consistency checks for the gauge structuresSU (3) and SU (2) × U (1), including commutator closure, Casimir values, and charge reconstruction. • Dictionary construction: A revised state dictionary that provides an exact cover of the 48 fermionic degrees of freedom of the Standard Model within the 60-state system. The bosonic sector is shown to be structurally consistent but is not yet claimed to be fully derived at theorem level. The manuscript also reports numerically close golden-ratio relations for the Cabibbo angle and the weak mixing angle, θC = arctan (1/φ³) sin²θW = 3/ (8φ) These relations are presented explicitly as phenomenological ansätze, not as first-principles derivations from the Standard Model. The purpose of this paper is therefore not to claim a completed unified theory, but to clearly define the claim boundary of the H₄/GZ60 framework: which components are mathematically exact, which are supported by algebraic audits, and which remain phenomenological or open problems. Topics include: • H₄ (600-cell) geometry• discrete cyclic algebraic systems• Standard Model fermion degree-of-freedom structure• SU (3) and electroweak gauge algebra audits• hydrogenic SO (4) symmetry• golden-ratio relations in physical parameters