Single-Field Inflation from the Rendering Algebra: Complete Derivation via Icosahedral Geometry

Abstract

We derive the complete primordial inflation programme from the rendering algebra R12 ∼= M3(C) ⊗ M4(C) with zero free parameters. Twenty-four results are established across six domains. Amplitude: the scalar power spectrum As = exp(−20 + 11Z/60) (0.06%), where 60 = |A5| is the order of the icosahedral rotation group. Geometry: the icosahedron is the unique Platonic solid with V=N=12 vertices and φ-coordinates; its combinatorial data (V, E, F, χ) = (12, 30, 20, 2) map isomorphically onto (N, f(1)NcTw, L/rank, Tw). Dynamics: inflation is proven necessary (Clifford grade hierarchy), Ne = 16π e-folds with NLO = 16π + αs (0.0004% cross-identity), and spectral running dns/d ln k = −(ns−1) 2 . Non-Gaussianity: f sq NL = 5(1−ns)/12 (singleclock theorem with 5/12 = degico /V) and f equil NL = 5(ns−1) 2/12. Isocurvature: Piso/Padi = (15/8) exp(−20/3) = 0.0025. Two new N=12 uniqueness theorems are proven. No inflaton field, no potential, and no slow-roll parameters are employed.