A Discrete Modular Algebra Framework Unifying the Standard Model and Periodic Law: Triple-Path Verification of the Z₅ Phase Generator and Complete Derivation of 118 Elements via Z₆₀ State Space

This paper presents a novel discrete modular algebra framework that conceptually unifies the shell structure of the periodic law and the discrete sector assignments of the Standard Model. By jointly analyzing a 24×24 Fibonacci-mod-9 multiplication table (V₂₄) and a 60-state cyclic space (Z₆₀) derived from finite shell capacities (2, 8, 18, 32), we demonstrate that these two fundamental structures are governed by the exact same algebraic engine. Key Findings and Contributions: Triple-Path Verification: We prove that the same Z₅ cyclic generator (step +2) arises independently from three distinct domains: (1) the physical Madelung orbital-filling sequence, (2) the modular coordinate congruence on Z₆₀, and (3) the geometric action of the H₄ / 600-cell rotation group. Complete Derivation of 118 Elements: By combining the Z₆₀ block structure with the Pauli exclusion principle and the Madelung ordering, the framework perfectly yields the complete known periodic table up to Oganesson (Z=118) without any ad-hoc parameters or empirical adjustments. Gauge-Higgs Conjugate Pair Candidate: The H₄ rotation action on the 60 axes yields an exact 5, 5, 25, 25 orbit decomposition. The two pure size-5 fixed-point orbits are structurally identified as the gauge-sector and Higgs-sector conjugate pair.