Paper 30 — High-Precision Quasi-Normal Mode Spectrum of the Black Circle Interior

We compute the quasi-normal mode (QNM) spectrum of the Black Circle interior — a de Sitter quantum geometry arising from the sedenion zero-divisor algebra — at 50 decimal places using mpmath arbitrary-precision arithmetic.
The physical damped mode under Candidate (a) boundary conditions is determined to be ω* = 1.92525087091...−0.41596770795...i (residual |H|=6.68×10⁻⁵¹), correcting an error in Paper 17 caused by a poorly chosen starting guess. The ratio |ω*|/|ω_Sch| = 1.9967±0.003 is compatible with the Frobenius factor 2 predicted algebraically in Paper 10 (Conjecture 6.3). A systematic scan of the upper half-plane Im(ω)>0 reveals two growing modes for l=2, m²=0 with timescales τ≈0.26–0.35 r_s/c, closing OP 18.5. These are four times shorter than the growing modes at l=0,1, establishing that the quasi-universality of growth timescales is specific to l=0,1 and fails at l=2.

Part of the series Zero-Divisors in Cayley–Dickson Algebras (Papers 1–32, Zenodo).

Papers referenced in this work:
Paper 10 — Loop, Threshold, and Residue (Frobenius factor prediction):
https://doi.org/10.5281/zenodo.20507204

Paper 13 — Quasi-Normal Modes of the Black Circle Interior (errata):
https://doi.org/10.5281/zenodo.20509979

Paper 14 — Black Circle Hypergeometric Reduction and Cavity Bound:
https://doi.org/10.5281/zenodo.20553409

Paper 15 — The Kummer Connection Matrix for the Black Circle:
https://doi.org/10.5281/zenodo.20554392

Paper 16 — The Limit-Circle Obstacle for Black Circle QNMs:
https://doi.org/10.5281/zenodo.20554795

Paper 17 — The Physical Boundary Condition at r=0 (errata):
https://doi.org/10.5281/zenodo.20555042

Paper 18 — Vacuum Instability and Majorana Spinors of the Black Circle:
https://doi.org/10.5281/zenodo.20573890