Paper 16 — The Limit-Circle Obstacle for Black Circle Quasi-Normal Modes

This paper identifies the precise obstacle to the quasi-normal mode connection problem for l≥1 in the Black Circle interior.

For all l≥1, r=0 is a regular singular point of the Regge–Wheeler chi-ODE with complex Frobenius exponents ϱ = (1±2iδ_l)/2, where δ_l = √(4l(l+1)−1)/2. Both Frobenius solutions are L²(0,ε) — a limit-circle singularity in Weyl's classification — requiring a self-adjoint extension to select the physical domain.

The s-homotopic substitution on the ρ₂ Frobenius branch transforms the w-ODE to the Gauss hypergeometric equation with parameters c̃ = 1−iδ_l, ã = −1/4+i(Ω−δ_l/2), b̃ = 5/4+i(Ω−δ_l/2), verified by a three-step algebraic proof.

The Kummer connection formula yields the exact coefficient K̃₁^(ρ₂,l) for all l≥1, reducing to the l=0 formula of Paper 15 when δ_l → i/2.

The physical particular solution w_phys satisfies the nonhomogeneous identity L_w[w_phys] = 2l(l+1)/z, proved algebraically and verified numerically to 8 decimal places — closing OP 15.1 negatively: no decomposition w_phys = Au_{ρ₁} + Bu_{ρ₂} exists. Physical masses for Levels 4 and 5 are derived explicitly: m²₄ = 4√(6x/π), m²₅ = 6√(6x/π), both below the Type II threshold under the cavity bound.