How Many Modes Determine a Star? Spectral Sufficiency Bounds for Asteroseismology via the Latent Theorem

Asteroseismology determines stellar interior structure from observed oscillation frequencies — eigenvalues of the stellar structure operator. Space missions (Kepler, TESS, PLATO) measure hundreds of oscillation modes for individual stars, yet no theory predicts how many modes suffice to constrain the density profile \(\rho(r)\) to a given accuracy \(\varepsilon\). We apply the Latent Sufficiency Theorem to the Sturm-Liouville eigenvalue problem governing stellar oscillations and derive an explicit bound: the number of modes needed is

\[N^* = \Theta\!\left(\frac{\log(1/\varepsilon)}{\log \rho_\star}\right)\]

where \(\rho_\star\) is the analyticity parameter of the stellar structure — a single number determined by the smoothness of the Brunt-Väisälä frequency profile \(N^2(r)\) and the sound speed \(c_s(r)\). For solar-type stars with smooth radiative interiors, \(\rho_\star \approx 3\)–\(5\), implying \(N^ \approx 15\)–\(25\) modes for 1% accuracy. For evolved red giants with sharp composition gradients (small \(\rho_\star\)), the bound correctly predicts the observed need for \(N^ > 50\) mixed modes. The framework provides the first principled answer to a foundational question in asteroseismology: when has enough been measured?