CMB and Temperature from the Spiral Theory Theory-Derived T(β), Last-Scattering Depth, and Closure

We derive a parameter–free temperature–depth law for the early universe and black holehorizons from a single algebraic axiom, Φ2 = Φ + 1, with roots φ, ψ, and the spiral timeequation tβ = tP φβ. Without using any observed temperature, the theory fixes the shapeT(β) = T0 φ−βby the limits β → 0 (singularity, infinite temperature) and β → ∞ (classical, T → 0) andby the requirement tβ+1/tβ = φ. A single calibration T∗ = T(βLSS) then determines thelast–scattering depth βLSS = ln(T0/T∗)/ ln φ and its coupling C(kLSS) = 2/L2kLSS . On thelate–time side, the dominant black-hole remnant population at N = F12 = 144 has Hawkingtemperature TH ∝ φ−N; identifying this with the bath in thermal contact yields the CMBtemperature scale TCMB ≈ 2.73 K as a derived quantity, not a fit parameter. Thus a singlespiral law T(β) ∝ φ−βsimultaneously controls (i) recombination depth, (ii) the present CMBtemperature, and (iii) the Hawking spectrum of black holes. We summarise how this closesThe CMB “temperature thread” in the main Asymmetric Spiral Theory paper. The angularpower spectrum Cℓ is fixed by the same derivation: from the derived primitives φ, βLSS,C(k) = 2/L2k, χ∗ (algebraic and to high precision) the line-of-sight integral yields Cℓ in onepass. All formulas lie in Q(φ); the only free element is the overall scale T0 (e.g., TP or oneobservational calibration).