The Physical Boundary Condition at R=0 for The Black Circle

This paper determines the physical boundary condition at r=0 for the Black Circle quasi-normal mode problem for l≥1, left open by Paper 16.

Three candidates are analysed: (c) norm conservation via the IIB-II conjecture — the oscillating cross-term 2Re(C₁C̄₂·r^{2iδ_l}) forces C₁C₂=0; (b) Hartle–Hawking condition — outgoing Klein–Gordon flux J^r(χ₂)|_{r→0} = +2δ_l requires C₂=0; (a) global dS₄ regularity — the homogeneous contributions φ_j^hom ~ r^{-1/2} are singular at r=0, forcing C₁=C₂=0. The hierarchy (a)⊂(b)⊂(c) is established.

The QNM condition of Paper 16 is corrected by the addition of the particular-solution coefficient A_part. Under Candidate (a), the first l=1 QNM spectrum is derived: a damped mode ω*=1.2979−0.0076i (τ=131 r_s/c) and a growing mode ω*=1.6449+0.7372i (τ=1.357 r_s/c), quasi-universal with the l=0 growing mode to 0.33%.

The exact threshold m²_thr=0.4605600022 is established (OP 17.3). All Paper 13 l≥1 QNM values are identified as extraction artifacts under Candidate (a). Candidates (a) and (b) give distinguishable predictions differing by more than 3 units in Re(ω), constituting the first experimentally falsifiable prediction of the series.

Part of the Cayley–Dickson Zero-Divisor Series (Paper 17).