Paper 27 — Two-Loop Induced Gravity and Branch Selection for the Black Circle Self-Consistency Equation

Paper 27 of the Cayley–Dickson Zero-Divisor Series. Companion to Paper 26 (doi:10.5281/zenodo.20693540).

This paper resolves four open problems left by Paper 26, all requiring explicit computation rather than new algebraic ideas.

In §1 we close OP 26.3 by deriving the exact ratio of the tachyonic vacuum-decay amplitude Im(W₁-loop)|_tach (Paper 18) to the Level-1 imaginary part Im(G_N⁻¹)|_{L₁} (Paper 26): the ratio equals 744.22/7 ≈ 106.32, a pure number independent of all parameters, identified as the Black Circle spacetime geometric integration factor. In §2 we close OP 26.5: numerical analysis of the self-consistency equation from Paper 26, Theorem 4.6 shows that the large branch G_N^(2) violates the cavity bound of Paper 14 by factors of 234× to 13,468×, while G_N^(1) is the unique physical branch for λ < λ_c^(SC) = 0.07531 (at M = 0.1, Λ = 1 in Planck units).

In §3 we close OP 25.2 by assembling the complete two-loop β-function for S₈₄^(f), applying the Machacek–Vaughn formula to the Yukawa matrices Y_j = γ_{s_j} ∈ M₈(ℝ) with invariants à = 16, B̃ = 32 derived from the Clifford algebra Cl(0,7): β^(2-loop)(λ) = (63λ²+4λ−16)/π² + (−189λ³−12λ+48)/(4π⁴)

The two-loop IR zero shifts by −2.0% to λ_IR^(2) = 0.46373. In §4 we close OP 26.6: substituting the two-loop running coupling into the Frolov–Fursaev formula gives the two-loop induced G_N, with perturbative correction 5%–13% at T = 0.1.

In §5 an unexpected algebraic result emerges: under Conjecture A (Black Circle mass M equals its spinorial excitation mass m_{4_s}), the self-consistency system reduces to a unique closed-form solution G_N^(A)(λ) = exp((1/K(λ) − P₀(λ))/164), determining both M and G_N algebraically for each λ. The cavity-physical window is λ <  λ_c^(A) = 0.01197.