Revised Claim Boundary for the H4/GZ60 Discrete Algebraic Framework

This manuscript presents a calibrated statement of the H4/GZ60 discrete algebraic framework, a structural proposal connecting 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 organized through the factorization60 = 12 × 5interpreted as a Z₁₂ × Z₅ cyclic structure. Within this setting, the manuscript separates results into three levels of claim strength: theorem-level results, exact algebraic audits, and phenomenological relations.The strongest result recorded here is an exact discrete SO(4) Casimir derivation reproducing the hydrogenic energy spectrumEₙ = −1/(2n²)together with the associated shell capacities in the adopted representation reading.At the algebraic level, the manuscript records exact generator-level audits for the gauge structuresSU(3) and SU(2) × U(1),including commutator closure, Casimir values, adjoint structure, and charge reconstruction.A revised state dictionary is introduced 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 coherent but is not yet claimed to be theorem-level closed.The manuscript also records numerically close golden-ratio relations for the Cabibbo angle and the weak mixing angle,θ_C = arctan(1/φ³)sin²θ_W = 3/(8φ)but these relations are explicitly presented 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 structurally supported by algebraic audits, and which remain phenomenological or open problems.Topics addressed include:・discrete algebraic structures in particle physics・H4 (600-cell) geometry・cyclic systems and symmetry structures・Standard Model fermion degree-of-freedom bookkeeping・hydrogenic SO(4) symmetry・golden-ratio relations in physical parameters