The Rendering Algebra and the Riemann Zeta Function: γ1 from B2=I, Axiom Reduction, Binary Allocation, and the Cauchy–Euler–Dirichlet Correspondence

Abstract

Two independent top-down paths within Array Cosmology determine the rendering tension γ1—the imaginary part of the first nontrivial Riemann zeta zero. Path A (Paper L, eikonal + Casimir ratio) gives γ1=70π/(9√
3) at 0.19%. Path B derives γ1=|Z2|/(π−3)=2/(π−3) at 0.07% as a theorem: B2=I generates Z2, whose information capacity log2
|Z2|=1 bit equals the area quantum ∆A=4 ln 2 (Paper XXIX) per KMS cycle TwZ=2Z=∆˜ωI (Paper XXIII). No physical interpretation labels are required—every step is algebraic or information-theoretic. Self-consistency of the two paths, combined with the NLO dimension rule at d=2 (horizon), yields γ1=2/(π−3)+Z 2 at 4.7 ppm with zero free parameters. This demotes γ1 from axiom to derived quantity: {12, π, γ1, ln 2, φ} → {12, π, ln 2, φ}.
The Cauchy functional equation f(x+y)=f(x)f(y), which governs confinement versus deconfinement in AC, is identified with the Euler–Dirichlet duality of ζ(s) in four physical realisations. At the confinement crossover βc= ln 2=β(p=2), every Boltzmann
weight becomes an exact power of two (e−λβc=2−λ), and the degeneracy recursion d(λ)=2λ−d(λ−4) follows from the tensor-product structure of R12 together with the Mihailescu identity N2 c −1=2Nc and the electroweak identity N2
f =2Nf . The partition function evaluates to ZR12 (βc)=4−2 1−Nf=31/8 exactly. Twenty-one results; axioms {12, π, ln 2, φ}; zero free parameters.