The Latent: Finite Sufficient Representations of Smooth Systems
Abstract
We define the Latent of a smooth system as the basis-free element of a graded Hilbert tensor algebra that completely characterizes the system's distributional, dynamic, and functional properties. A single dimensionless number \(\rho\) — the system's Latent Number — measures its intrinsic compressibility.
Main theorems. The Latent Theorem (Theorem 1, §3.2) proves that every Latent with \(\rho > 1\) in some extractable basis admits a finite approximation of size \(N = \Theta(\log(1/\varepsilon)/\log\rho)\) per mode, independent of ambient dimension and basis once an effective-rank reduction is fixed. The Extended Latent Theorem (Theorem 2, §3.3) extends this to non-smooth systems via sufficient smooth representations. The Rational Latent Theorem (Theorem 8, §4) predicts Padé convergence rates with the pole basis as constructive optimum. All three are basis-free: the COS expansion, eigenvalue conditioning, the Fokker–Planck generator, and Knowledge Artifacts are recovered as coordinate representations of the same underlying Latent in specific bases.
Analyticity–Rate Duality and controlled benchmark. The Bernstein-ellipse parameter that governs spectral convergence is the same \(\rho\) that governs the Cramér rate of importance sampling. We validate this duality on a controlled Gaussian-portfolio benchmark (§3.6): at \(\rho \approx 6\) we measure a variance-reduction ratio of \(1.7 \times 10^4\) against naive Monte Carlo (95% CI \([1.3, 2.1] \times 10^4\)), and at \(\rho \to 1\) (Student-\(t\) tails) we observe IS failure in the predicted form.
Higher-grade extraction. The three-body problem (§8.4) demonstrates the grade hierarchy beyond pairwise methods: grade-3 co-skewness is \(30\)–\(600\times\) larger for chaotic orbits than periodic ones, the kinematic rank is universally 4 (confirming the rank bound), and the Fourier-vs-Padé extraction gap is directly measurable. The parametric map from initial conditions to Latent is itself a smooth 4-dimensional manifold — the meta-Latent — yielding a finite representation of orbit families, not just individual orbits.
Algebra and structural consequences. The basis-free formulation makes visible: (i) an optimal extraction problem — the basis maximizing \(\rho\) minimizes coordinate count; (ii) higher-order entanglement — grade-\(r\) Latents for \(r \geq 3\) carry irreducible multi-body interactions no pairwise decomposition can represent; (iii) Latent transfer — a Latent extracted in one basis can be re-expressed in another without re-extraction. The Latent Algebra (Proposition 7, §6) is the universal graded contraction algebra generated by the underlying Hilbert space.
Further applications. The framework is applied to la
Novelty
Unifying spectral representations across finance, physics, ML, and geometry under a single graded Hilbert tensor algebra object ('the Latent') with a basis-free finite-sufficiency guarantee governed by a single compressibility parameter ρ, plus the Analyticity–Rate Duality connecting Bernstein ellipse convergence to Cramér importance-sampling efficiency.