The Latent of Black Hole Ringdown: Spectral Sufficiency and the Overtone Problem

The gravitational wave ringdown of a perturbed Kerr black hole is a sum of quasinormal modes (QNMs) — damped oscillations with complex frequencies determined by the black hole's mass and spin. We show that the Latent representation theorem provides a principled bound on spectral sufficiency: the number of QNM overtones required for fractional accuracy ε in the ringdown signal is \(N^* = \Theta(\log(1/\varepsilon) / \log \rho)\), where the analyticity parameter \(\rho = |\mathrm{Im}(\omega_1)| / |\mathrm{Im}(\omega_0)|\) is the damping rate ratio of the first overtone to the fundamental mode.

Numerical computation with the full Kerr QNM spectrum reveals that \(\rho \approx 3\) is a topological universal: it varies by less than 3% across all spins \(0 \leq \chi < 1\), ranging from \(\rho = 3.08\) (Schwarzschild) to \(\rho = 3.00\) (near-extremal Kerr). The origin is the WKB structure \(|\omega_{I,n}| \propto (n+1/2)\), which gives the exact ratio \(\rho = 3\) in the eikonal limit. This universality implies a spin-independent sufficiency bound: \(N^* \approx 5\) for 1% accuracy regardless of black hole spin.

The grade decomposition of black hole perturbation theory assigns grade \(k\) to the \(k\)-th overtone, with amplitudes bounded by \(|A^{(k)}| \leq C\rho^{-k}\). This explains the observed hierarchy in which the fundamental mode dominates: it is the grade-1 component of the perturbed Kerr geometry, and higher overtones are exponentially suppressed higher-grade corrections.

We establish a connection between QNM pole structure and the Padé–Stieltjes pipeline: the retarded Green's function of the Teukolsky equation is meromorphic in the lower half-plane with poles at QNM frequencies, and Leaver's continued fraction method (1985) for computing QNMs is mathematically equivalent to the Padé approximation of this Green's function. The non-meromorphic branch cut contribution (Price tail, \(\sim t^{-(2\ell+3)}\)) defines a crossover time \(t_{\mathrm{tail}}\) beyond which the QNM representation breaks down — the Latent framework quantifies this boundary.

For GW150914-like events at current LIGO sensitivity, the framework predicts \(N^*_{\mathrm{det}} = 2\) statistically significant overtones, consistent with the analyses of Isi et al. (2019, 2021) and resolving the apparent tension with Giesler et al. (2019) who fit \(N = 7\) overtones to the same data. The extremal Kerr limit \(\chi \to 1\) constitutes a spectral phase transition — not through \(\rho\) collapsing, but through all damping rates vanishing (\(\tau_n \to \infty\)): the black hole becomes a perfect resonator, with the QNM representation remaining compressible (\(\rho \approx 3\)) but the individual modes becoming quasi-bound states.