What Trace Powers Reveal About Log-Determinants: Closed-Form Estimators, Certificates, and Failure Modes

Computing logdet(A) for large symmetric positive definite matrices arises in Gaussian process inference and Bayesian model comparison. Standard methods combine matrix-vector products with polynomial approximations. We study a different model: access to trace powers pk = tr(Ak), natural when matrix powers are available. Classical moment-based approximations Taylor-expand log(λ) around the arithmetic mean. This requires |λ - AM| < AM and diverges when kappa > 4. We work instead with the moment-generating function M(t) = E[Xt] for normalized eigenvalues X = λ/AM. Since M'(0) = E[log X], the log-determinant becomes logdet(A) = n(log AM + M'(0)) -- the problem reduces to estimating a derivative at t = 0. Trace powers give M(k) at positive integers, but interpolating M(t) directly is ill-conditioned due to exponential growth. The transform K(t) = log M(t) compresses this range. Normalization by AM ensures K(0) = K(1) = 0. With these anchors fixed, we interpolate K through m+1 consecutive integers and differentiate to estimate K'(0). However, this local interpolation cannot capture arbitrary spectral features. We prove a fundamental limit: no continuous estimator using finitely many positive moments can be uniformly accurate over unbounded conditioning. Positive moments downweight the spectral tail; K'(0) = E[log X] is tail-sensitive. This motivates guaranteed bounds. From the same traces we derive upper bounds on (det A)1/n. Given a spectral floor r ≤ λmin, we obtain moment-constrained lower bounds, yielding a provable interval for logdet(A). A gap diagnostic indicates when to trust the point estimate and when to report bounds. All estimators and bounds cost O(m), independent of n. For m ∈ 4, ldots, 8, this is effectively constant time.