The Barbero–Immirzi Parameter, Black-Hole Entropy, and the Area Gap from the R12 Rendering Algebra: McKayFolding, the SU(2)4 Horizon Category, and the Complete Entropy Expansion

Abstract

The Barbero–Immirzi parameter is derived as γBI = ln2/(π√3) = 0.127384 from the rendering algebra R12 = M3(C)⊗ M4(C) of Array Cosmology, as the exact solution of the Domagała–Lewandowski normalization restricted to the puncture spectrum the algebra admits. The restriction is a theorem: the rotation module of M4(C) ∼=Cl(3,1)⊗Cis 1 2 ⊕ 1 2 with Casimir 3 4, so horizon punctures carry spin 1 2 exclusively.