Latent Complexity: A Computable Theory of System Difficulty for Smooth Systems

We construct a complexity theory for smooth (analytic) systems based on the Latent framework. We define three orthogonal complexity axes: the Latent Number \(\rho\) (description complexity: how many numbers are needed to represent the system), the effective grade \(r^\) (interaction complexity: the highest order of irreducible coupling present), and the extraction cost \(\mathcal{C}_E\) (computational complexity: the cost of recovering the Latent from observations). We introduce four complexity classes — L-ENTIRE (\(\rho = \infty\)), L-ANALYTIC(\(\rho\)) (\(\rho > 1\)), L-BOUNDARY (\(\rho = 1\), \(\rho^ > 1\)), and L-SINGULAR (\(\rho^ = 1\)) — and prove they form a strict hierarchy (Theorem 1). We prove the Description–Computation Theorem (Theorem 2): for any system in L-ANALYTIC(\(\rho\)), every bounded linear query is answerable in \(O(N^) = O(\log(1/\varepsilon)/\log\rho)\) operations after a one-time extraction costing \(\mathcal{C}_E\), establishing that description complexity upper-bounds query complexity for smooth systems. We prove the Grade–Depth Correspondence (Theorem 3): the effective grade of a system's Latent is isomorphic to the depth of a natural "interaction circuit" model, where grade-\(r\) gates compute \(r\)-linear contractions on Hilbert space. We prove the Computability Theorem (Theorem 4): the Latent complexity class of a specific smooth system is decidable — \(\rho\) is computable from finite data, grade is measurable, and extraction cost is bounded by the system's dimensionality — in contrast to Kolmogorov complexity (uncomputable) and classical complexity classes (membership unresolved for most problems). We establish formal relationships between Latent complexity and seven classical measures: Kolmogorov complexity, Shannon entropy, VC dimension, Rademacher complexity, Kolmogorov \(n\)-widths, communication complexity, and circuit complexity. We show that Latent complexity is strictly more informative than any single classical measure for smooth systems: it provides exponential rates where classical measures give only asymptotic classes, it is computable where Kolmogorov complexity is not, and it captures interaction structure where entropy-based measures are blind. We demonstrate the theory on systems from financial risk (grade-1, \(\rho \approx 1.3\): L-ANALYTIC), turbulence (grade-2 with \(\rho \to 1\) as \(\text{Re} \to \infty\): approaching L-SINGULAR), neural networks (grade-1, \(\rho\) from data covariance: L-ANALYTIC with computable model size), and the three-body problem (grade-3, \(\rho\) orbit-dependent: L-ANALYTIC for periodic orbits, approaching L-BOUNDARY for chaotic ones). The main theorems are formally verified.

Keywords: complexity theory, Latent Number, analyticity, s