Paper 15 — The Kummer Connection Matrix for the Black Circle

This paper derives the exact Kummer connection matrix for the Black Circle interior quasi-normal modes.

Starting from the Papperitz–Riemann classification of the interior Regge–Wheeler ODE as a Gauss hypergeometric equation with three regular singular points (established for all angular momenta l and masses m²), the Kummer connection formula yields the exact coefficient D_in^true = K₁·2^{iΩ} relating the growing Frobenius mode at the Eddington–Finkelstein junction to the interior solution.

For l=0 the Legendre cascade gives K₁ = 2^{κ−1}/(1+κ) in closed form; the true growing mode satisfies (2e)^κ = 2(1+κ), giving κ_grow = 0.73472367 (τ = 1.361 r_s/c) via Lambert W. No damped l=0 modes exist at m²=0 (algebraic proof).

A Type II l=0 curve of damped oscillatory modes is found for m² > m²_thr ≈ 0.464, constituting a new prediction absent in the approximate condition. Open problem OP 15.1 (connection formula for l≥1) is carried forward to Paper 16.