The Subgroup Ladder is the Chinese Remainder Theorem: Base 60 as the Canonical Coordinate of 2I

This paper establishes that the Chinese Remainder decomposition Z/60Z ≅ Z/3Z × Z/4Z × Z/5Z coincides with the successive coset decomposition along the unique maximal subgroup chain {e} ⊂ Z₃ ⊂ A₄ ⊂ A₅ of the icosahedral group. The indices 3, 4, 5 serve simultaneously as CRT moduli and as geometric symmetry layers: triangular, tetrahedral, and pentagonal. This identification yields a canonical base-60 coordinate on A₅ via CRT reconstruction, extending to a Z/2Z × Z/60Z coordinate on the binary icosahedral group 2I through its double cover. The coordinate map is not a group homomorphism; its failure is measured by a 2-cocycle σ: A₅ × A₅ → Z/60Z encoding the non-abelian multiplication of A₅ within the abelian numeration system. Applied to the icosian ring, the construction provides an intrinsic base-60 numeration of E₈ in which lattice addition and unit multiplication are coupled through σ. As a consequence, the E₈ → 2I branching coefficients decompose along the same three CRT factors, resolving the last open entry in a previously established correspondence between Poincaré Dodecahedral Space and exceptional geometry.