The Cosmological Energy Budget, Gauge Couplings, and CKM Matrix from R12 Structural Constants

Abstract

We derive the complete cosmological energy budget, all three Standard Model gauge couplings, and the CKM quark mixing matrix from the five structural constants {N = 12, C2 = 3, Tw = 2, (π − 3), ln 2} of the rendering algebra R12 ∼= M12(C).
Part I (Cosmology): The baryonic density Ωb = Z/Tw (0.15%), dark-to-baryon ratio ΩDM/Ωb = 12π/7 (0.27%), and dark energy ΩΛ ≈ b0Z (0.34%) are all proportional to the vacuum friction Z = (π − 3) ln 2.
Part II (Gauge Couplings): The fine-structure constant inverse α −1 = N2 − (π − 3)−1 + Z = 137.036 (0.0003%) follows from spectral mode counting on M12; the Weinberg angle sin2 θW = ln 2/3 (0.07%) from additive Cauchy duality and Landauer equipartition over dim(M3) = 9; and the strong coupling αs = ζ(3)Z (0.06%) from the 3D Bose–Einstein thermal density normalised by the KMS temperature Tw.
Part III (CKM Matrix): The Cabibbo element |Vus| = e−3/2(1.0%), the bottomcoupling |Vcb| = e−(3+2Z) (0.2%), and |Vub| = |Vus| |Vcb|×5/12 (0.4%) emerge from Casimirbarrier tunnelling with QCD channel blocking.

The Wolfenstein parameter A = e −2Z is an exact identity, and δCKM = ζ(3) rad (0.5%) reproduces the Jarlskog invariant J = 3.15×10−5 (0.8%). A cross-sector theorem |Vus| 2 × (ms/md) = 1 links CKM mixing to the quark mass hierarchy.