The Latent Path Integral

We prove that a single saddle-point formula governs quantitative approximation in every system whose action functional admits a Latent grade decomposition. Let \(\mathcal{A} = \mathcal{A}_2 + \mathcal{A}_{\geq 3}\) be a functional on a path space \(\Gamma\), where \(\mathcal{A}_2\) is quadratic (grade 2) and \(\mathcal{A}_{\geq 3}\) collects all terms of grade 3 and above. If the Latent number \(\rho\) — a dimensionless parameter measuring the dominance of grade-2 over higher grades — satisfies \(\rho > 1\), then the path integral factorizes as

\[F = F_{\text{Gauss}} \cdot \bigl(1 + \delta\bigr), \qquad |\delta| \leq \frac{2C}{\rho^3},\]

where \(F_{\text{Gauss}} = K \cdot (\det \operatorname{Hess} \mathcal{A}_2)^{-1/2}\) is the Gaussian (grade-2) approximation and \(C\) is a grade-structure constant independent of \(\rho\).

This formula is domain-independent. We exhibit twelve concrete instantiations where the abstract \(\rho\) acquires a physical or mathematical identity. The domains group into three mapping types:

1. Direct mappings (larger parameter \(\to\) cubically better approximation): the COS option pricing strip width, the Euler product prime power, the Nash equilibrium payoff curvature, the Bayesian sample size, the statistical mechanics system size, the stochastic control authority, the large deviations event count, and the distance from the renormalization group fixed point. 2. Inverse mappings (larger parameter \(\to\) cubically worse approximation): the Navier-Stokes Reynolds number, the machine learning parameter-to-data ratio, and the quantum circuit depth. 3. Parametric mappings (shifting the entire action rather than just higher-order terms): the rough volatility exponent \(\alpha = H - 1/2\).

The cubic decay \(O(\rho^{-3})\) is universal across all direct and inverse domains; rough volatility provides a parametric bridge where the correction is first-order. The large deviations bridge identifies the Latent Path Integral as the generalization of the saddlepoint approximation to non-exponential families. The renormalization group bridge provides the meta-explanation: grade-2 dominance near critical points is a consequence of RG universality — irrelevant operators decay as powers of the distance to the fixed point, and the leading irrelevant correction is grade 3. All 110 theorems are formally verified in the Lean 4.

Keywords: path integral, saddle-point approximation, grade decomposition, Latent number, universality

MSC 2020: 41A60, 81S40, 60H30, 11M06, 91A10