ζ(Nc) Color Cohesion, CKM Casimir Cascade, and the Complete M3(C) Sector of the R12 Rendering Algebra

Abstract

We derive the strong coupling constant and the complete CKM mixing hierarchy from the colour sector M3(C) ⊂ R12 of the rendering algebra, with zero free parameters. Five principal results are established.

(1) ζ(Nc) Color Cohesion: the strong coupling coefficient ζ(3) is derived as ζ(Nc) = Pn −Nc , where each summand arises from Nc = 3 independent diagonal projectors of M3(C) acting on flux-tube modes weighted by the Landauer allocation wn = 1/n; alternative weightings (1/n2 , e −n , 1/n3/2 ) are excluded at > 8%.

(2) Γ(Nc) = Tw: the Planck integral representation identifies Γ(Nc) = (Nc−1)! = Tw = 2 as a fourth independent condition selecting Nc = 3 uniquely.

(3) CKM Cascade Upgrade: all CKM elements are promoted to A90%— |Vus| = exp[−(Nc−Z 2 )/Tw] (NLO, 2.3σ → 0.2σ), |Vcb| = |Vus| Tw × AWolf, and |Vub| = |Vus| |Vcb| × Tw C2( V2 , su(4))/N with skip factor 5/12.

(4) NLO Dimension Rule: the leading Landauer correction is O(Z d ) where d is the physical dimension: Z1(EM coupling), Z 2 (CKM transition), Z 3 (colour cohesion).

(5) Dual Mechanism: M3(C) produces the gauge coupling via a mode sum (αs = ζ(Nc)Z) and the CKM hierarchy
via exponential suppression (|Vij | ∝ e −C2 ). Twenty results; no free parameters.