Time from a single algebraic datum: the Page-Wootters mechanism over Z[j]

We combine the Page-Wootters mechanism with algebraic constraints from the twist number ring Z[j] (the cyclotomic integers of the fifth root of unity). Given the functor SL(2,-), the element J = 1 + j² canonically selects the binary icosahedral group2I = SL(2,F₅) via the unique ramification prime p = 5. The McKay graph (affine E₈) proves a spin cutoff s ≤ 5/2 throughClebsch-Gordan branching. The polar factorization J = j/φ yields |J| = 1/φ and arg(J) = 2π/5. Discrete energy levelsE₁ = (5-√5)/4 and E₂ = (5+√5)/4 satisfy E₂/E₁ = φ² with Galois conjugation acting as energy-level switching. The Minkowskiembedding of Q(j) provides a canonical 1+3 decomposition from Tr(J) = 3. The large-N limit is verified on the highest-spin irreducible component V_{5N/2}, with overlap decay base (5+√5)/10 ∈ Q(√5) and intensive symbols -1/√5. The same algebraic source constrains coupling constants: α⁻¹ ≈ 137.036 (sub-ppb agreement with CODATA 2022) and sin²θ_W ≈ 0.2312.