The Multi-Scale Derivation of Gauge Couplings, the Electroweak Hierarchy, and the Scattering Programme of the R12 Rendering Algebra

Abstract
We clarify the dynamical content of the R12 rendering algebra by establishing three results. First, the three Standard Model gauge couplings are derived at their respective natural scales—not at a single unification point: α−1
0 = 122−1/(π−3) ≈ 136.937 at the UV lattice scale (Paper III), with the observed α−1 = 137.036 obtained after Landauer IR descent + (π −3) ln 2 + Kknot/2; αs = ζ(3)Znoise ≈ 0.1180 at the Z-pole (Paper V), and sin2
θW = 3/8 as a scale-independent trace ratio (Paper XIX). Standard Model renormalisation-group evolution from these derived
values reproduces all three observed couplings at MZ to within 0.02–0.07%, with zero free parameters. This multi-scale structure differs fundamentally from Grand Unified Theories, which require all couplings to meet at a single energy; in AC, no
such meeting is claimed or required