The Latent Number ρ: A Universal Diagnostic for Computational Complexity

We introduce the \(\rho\)-diagnostic: a cross-domain methodology for assessing the computational complexity of smooth systems through a single computable parameter, the Latent Number \(\rho\) (the system's intrinsic compressibility; formally, the analyticity parameter). We prove the Universal Complexity Theorem: for any system admitting an analytic representation with analyticity parameter \(\rho > 1\), the number of degrees of freedom needed to achieve accuracy \(\varepsilon\) is \(N^(\varepsilon) = \Theta(\log(1/\varepsilon) / \log \rho)\), independent of ambient dimension, choice of basis, and computational domain. We show that \(\rho\) has natural, computable interpretations across six major scientific domains: as the Bernstein ellipse parameter in distribution theory, as the eigenvalue decay rate of data covariance in machine learning, as the spectral gap of the Fokker–Planck generator in dynamical systems, as the Lindblad spectral gap in quantum mechanics, as the grade decay rate in fluid dynamics, and as the analyticity strip width of the interaction potential in molecular simulation. We prove a Phase Transition Theorem: the boundary \(\rho = 1\) is a sharp computational phase transition — below it, finite spectral representation is impossible; above it, exponential convergence is guaranteed. We develop practical algorithms for computing \(\rho\) from data (empirical spectral decay fitting), from parametric models (closed-form expressions for standard distribution families), and from governing equations (spectral gap extraction). We establish the Analyticity–Rate Duality: \(\rho\) simultaneously governs the convergence rate of deterministic spectral methods (COS, Galerkin, Chebyshev) and the efficiency of optimal importance sampling for rare-event simulation, unifying two literatures that have developed independently. We demonstrate the diagnostic on systems from financial risk (portfolio loss: \(\rho \approx 1.1\)–\(3.0\), \(N^ \approx 16\)–\(145\)), neural network training (data covariance: \(\rho\) predicts optimal model size), turbulence (Kolmogorov cascade: \(\rho\) from viscous cutoff determines inertial range), plasma confinement (MHD generator: \(\rho\) predicts disruption time), and protein folding (conformational free energy: \(\rho\) from the Hessian eigenspectrum). The key results are formally verified in Lean 4.

Keywords: Latent Number, analyticity parameter, computational complexity, spectral methods, Monte Carlo, phase transition, formal verification

MSC 2020: 65M70, 65C05, 41A25, 47A10, 68Q25