Formal Analysis of QNM Spectral Ratio Universality in the Extremal Kerr Limit

We present a formal proof that the spectral ratio \(\rho = \gamma_1/\gamma_0\) of quasi-normal mode (QNM) damping rates remains universal in the extremal Kerr limit (\(\chi \to 1\)). While individual mode lifetimes \(\tau_n = 1/\gamma_n\) diverge as the fundamental damping rate \(\gamma_0 \to 0\), the ratio \(\rho \approx 3\) is preserved. This result establishes that the extremal limit constitutes a "resonator transition" — where the black hole becomes a perfect resonator with divergent quality factor \(Q = \omega_R/\gamma_0 \to \infty\) — rather than a phase transition in the spectral structure itself. The proof is machine-verified in the Platonic kernel with 33 theorems covering lifetime divergence, ratio independence, quality factor scaling, mode count stability, exponential decay dynamics, convergence of constant sequences, quality factor bounds, complex frequency structure, lifetime continuity, and mode summation properties. All results are exportable to Lean 4.