The Universal Padé–Stieltjes Machine: One Algebraic Pipeline from Lognormal Sums Through the Riemann Zeta Function to the Three-Body Problem

We identify a single algebraic pipeline — Moments → Padé–Stieltjes resummation → Rational characteristic function → Distribution/Trajectory — that solves three structurally different problems:

1. Lognormal sums (statistics/finance): \(S = \sum w_i e^{Y_i}\), \(Y \sim \mathcal{N}(\mu, \Sigma)\). Closed-form moments exist but the MGF Taylor series has zero convergence radius. Padé recovers the characteristic function as a rational function. Result: the first fully algebraic CDF formula for correlated lognormal sums.

2. Riemann zeta distribution (number theory): \(|ζ(1/2+it)|\) on the critical line. Empirical moments \(M_k = \frac{1}{T}\int_0^T |ζ(1/2+it)|^{2k}\,dt\) grow super-exponentially. The √-transformed Padé–Gil-Pelaez pipeline recovers the CDF to 0.45% max error — 100× more accurate than the Selberg lognormal model.

3. Three-body trajectories (celestial mechanics): Taylor coefficients of the ODE solution, resummed by Padé to extend beyond the convergence radius. Machine-precision (\(10^{-13}\)) orbits from 880 evaluations per period.

The three problems are unified by a single structural theorem: if the target function is the Laplace/moment-generating transform of a positive measure, the diagonal Padé approximant converges despite the divergent Taylor series (Baker–Graves-Morris, Theorem 5.4.2). We call this the Padé–Stieltjes machine.

We introduce the Two-Latent Decomposition: the Padé approximation captures the smooth Latent \(\mathcal{L}_{\rm smooth}\); the residual is the oscillatory Latent \(\mathcal{L}_{\rm osc}\), which for the zeta function is structured by the Riemann zeros and itself admits a finite representation. The unification \(\mathcal{L}_{\rm smooth} \oplus \mathcal{L}_{\rm osc} = \mathcal{L}_{\rm unified}\) closes the Latent algebra and connects to the Riemann Hypothesis: the oscillatory Latent's structure constrains the zero locations.