One-Loop Stability and Backreaction of The Black Circle

Twentieth paper in the Cayley–Dickson zero-divisor series.

Papers 11–19 developed the canonical quantisation, quasi-normal mode (QNM) machinery, and spinorial sector of the Black Circle — the spherically symmetric spacetime sourced by the 84-mode scalar action S₈₄ of the sedenion algebra 𝒜₄ under the IIB-II conjecture (norm N = algebraic correlate of gravity). Paper 18 established that the one-loop imaginary part of the effective action is positive (the condensate C₀ is metastable) but left open whether C₀ is a local minimum or a saddle point of the one-loop effective potential, and whether the gravitational backreaction under IIB-II affects this conclusion.

This paper answers both questions in three steps.

Step 1 [Derived]: the classical potential is V(C) = 64λC⁴; the Hessian at C₀ has 7 negative eigenvalues in the irr₇ sector (tachyonic saddle point), 14 zero eigenvalues (Level 2), and 63 positive eigenvalues (Levels 3–5).

Step 2 [Derived]: the Coleman–Weinberg direct mass correction in the irr₇ direction is positive and equal to 32λ²C₀²/(3π²) ≈ +1.08λ²C₀², insufficient to flip the sign of the tachyonic mass at weak coupling.

Step 3 [Derived, cond. IIB-II]: the self-consistent Einstein equation at one loop shifts the condensate by a factor (1 + 0.912λ); gravitational backreaction dominates the direct correction by a factor of 54, making the effective tachyonic mass more negative by a fractional amount 1.79λ at one loop — Case (a): metastability is enhanced. IIB is perturbatively consistent and does not require reformulation.

First IIB-II loop prediction [Conjecture, cond. IIB-II]: the one-loop expectation value of the gravitational correlate N is negative and proportional to −8.41λC₀². Quantum fluctuations of the S₈₄ fields reduce N below its classical value.

Four open problems are stated, including the two-loop RGE (OP 20.1) and the one-loop β-function β(λ) = 63λ²/π² > 0 (proposed for Paper 21).