Paper 13 — Quasi-Normal Modes of the Black Circle Interior

Paper 12 established that standard Wronskian-shooting is invalid for the Black Circle quasi-normal mode (QNM) problem: the Frobenius exponents at rₛ are α_in = ±iω/2 (de Sitter interior) and α_out = ±iω (Schwarzschild exterior), with exponent ratio α_out/α_in = 2 equal to the Frobenius algebraic ratio ‖Lₐ‖²_F|ₙ₌₄ / ‖Lₐ‖²_F|ₙ₌₃ = 2 from Paper 7. This paper closes Open Problem 12.6 in three stages and delivers the complete QNM spectrum of the Black Circle interior.

(i) Connection matrix — feasibility analysis. A naive 2×2 algebraic matrix M(ω) : (c₊, c₋) → (b₊, b₋) does not exist for ω ≠ 0: the four functions {ε^{±iω/2}, ε^{±iω}} are linearly independent over ℂ. The correct ingredient is a hypergeometric connection formula for the de Sitter Regge–Wheeler ODE (OP 13.1). An epistemic flag is raised: the condition [dχ/dr] = 0 at rₛ is not established from Barrabès–Israel alone and is carried as OP 13.2.

(ii) EF matching condition and interior recurrence. The regular interior solution is purely even: χ_in(r) = r^{l+1} Σ c_{2k} r^{2k}. A step-2 three-term recurrence with explicit coefficients αₙ, βₙ, γₙ is derived. The Frobenius exponent mismatch vanishes in tortoise coordinates — it is a coordinate artifact. The matched QNM condition reduces to a single purely interior equation: H_BC(ω) = D_in(ω) − 2^{iω/2} e^{iω} = 0, where D_in is the Frobenius coefficient of (1−r)^{−iω/2} extracted via an exact Wronskian formula. The exterior Schwarzschild geometry provides the radiation channel but does not enter the eigenvalue equation. A general EF matching formula for any two-domain metric is derived. Hawking equivalence is established: the Black Circle Hawking spectrum is identical to the Schwarzschild spectrum for the same mass M.

(iii) Complete QNM spectrum. The spectrum has two structurally distinct families. Type I (imaginary-axis, overdamped): twelve non-oscillatory modes, one per (level, l) entry, residuals < 5×10⁻¹⁵; longest-lived: Level 2, l=0, τ = 12.305 rₛ/c. Type II (oscillatory): four modes, all l=2, n=0, ratio |ω_BC/ω_Sch| = 2.168 ± 0.002 — the √2 prediction of Paper 10 is not confirmed. A BF instability growing mode is found for Level 1, l=0: ω_grow = +4.139i, e-folding time τ_grow = 0.242 rₛ/c — the first numerical determination of the Black Circle BF instability exponent. The cavity stability bound λGM² < 0.003 is approximately 225× more restrictive than the de Sitter BF bound of Paper 11. A linear mass law Im(ω_BC) ≈ −0.0847 − 0.0728 m² is identified for Type II modes. The Frobenius factor 2 is traced through a complete chain: sedenion algebra → metric geometry → ODE singularity → tortoise log-structure → EF matching phase → exact QNM frequencies.