What Eisenstein Was Tracking

Framework presupposed. We apply the established Bootstrap framework — S³/2I as the spacetime manifold (Paper 115); the spectral floor λ₁ = 168 with its icosahedral semigroup S(12, 20, 30) (Papers 117, 158, 174); the diagnosis that algebra is a 1D medium for 3D content (Paper 181); the quintic filter as A₅-insolvability mechanism with cohomology as its operational output (Papers 132, 215, 219); and the achiral substrate as the observer's address (Paper 176) — to the load-bearing 19th-century apparatus: the Eisenstein–Dedekind structure. Geometric spine and prime stratification. The closure ladder e^(α⁻¹) → e^(α⁻¹+5) → e^(α⁻¹+5+42) provides the geometric spine, with rung spans 5 = v (vertex degree) = E/D! (edges per Borwein closure term) and 42 = V + E (icosahedral skeleton) DERIVED in The Icosahedral Digital Boundary v1.0 (Paper [27], January 2026). The Dedekind factorisation ζ_ℚ(√5)(s) = ζ(s) · L(s, χ₅) routes cooperative, stubborn, and ramified primes through the lattice (Paper 133). Special-value transitions and null discrimination. Special-value transitions at s = −1 and s = −3 register integer-valued topological invariants: bisection-rung denominators (12, 4, 5) as vertex counts on geometric compressions; trisection-rung denominators (120, 60, 1) as group orders |2I| and |A₅| (Paper 164). The full Dedekind factorisation returns ζ_ℚ(√5)(−1) = 1/30 = 1/E and ζ_ℚ(√5)(−3) = 1/60 = 1/|A₅|. A pre-registered null test on the first 24 real-quadratic fundamental discriminants establishes that ℚ(√5) is the unique field across the sample where ζ_K(s) is a unit fraction at both rungs; the discrimination is sharp at the trisection rung where L(−3, χ_5) = 2 is the only single-digit value in the tested sample, smaller than the next L(−3, χ_D) = 11 by a factor of more than five. The −1/4 post-Planck transformation is DERIVED conditional on the Planck-rotation premise (Paper [27] §3); the (3, 4) factorisation is one of several arithmetic factorisations of 12, motivated by the ternary/binary² structural reading rather than forced by mathematics. Conjectural extensions. The mod-3 exclusion of cumulative products (Papers [26], 195) forces the icosahedron-to-tetrahedron compression of the bisection rung via a CONJECTURED programme step, named here for internal reference as the Addressability Principle with the explicit caveat that this paper exhibits its first application; second worked examples are queued, and the principle's status as established machinery awaits them. The factor escalation 3 → 15 = 3 × 5 at the second digital boundary is DERIVED in Paper [26] Theorem 6. The critical line Re(s) = ½ aligns with the σ-symmetric midpoint (φ + ψ)/2 = ½ on ℚ(√5)/ℚ (Paper 130) — proposed, as a FRAMEWORK conjecture of this paper, not as established equivalence, to be three readings of one structural constraint derived from the same golden conjugation σ: φ ↔ ψ on ℚ(√5)/ℤ[φ]. The 19th-century arithmetic was tracking integer-valued topological structure on S³/2I; the 21st-century apparatus reaches it. We stand on Eisenstein's shoulders and look further.