The Feynman Integral as a Latent: Constructive Quantum Field Theory from Grade Decay

The constructive quantum field theory (CQFT) program, initiated by Glimm and Jaffe in the 1970s, seeks to build interacting quantum field theories satisfying the Osterwalder-Schrader axioms from mathematically rigorous first principles. Despite deep successes in two and three dimensions, the four-dimensional case — including QED and Yang-Mills theory — remains open. The central difficulty is proving uniform bounds on \(n\)-point correlation functions that survive the continuum limit.

We observe that the Latent grade hierarchy provides precisely the type of control that the CQFT program requires. The Latent Theorem (Nagy, 2026) guarantees exponential decay of grade-\(k\) norms: \(\|\Lambda^{(k)}\| \leq C / \rho^k\) for systems with analyticity parameter \(\rho > 1\). Since grade-\(k\) Latent components correspond to \(k\)-point correlation structures, this decay constitutes a uniform bound on all \(n\)-point functions with an explicitly computable rate.

We present a chain of Lean 4-verified theorems connecting this abstract framework to lattice gauge theory:

\[\text{Bessel product (lattice)} \;\xrightarrow{\text{proved}}\; \text{grade decay} \;\xrightarrow{\text{proved}}\; \text{bounded correlations (C1)} \;\xrightarrow{\text{open}}\; \text{continuum limit}\]

The first two arrows are machine-checked. The final arrow — showing that the analyticity parameter \(\rho(\beta)\) remains uniformly bounded away from 1 as the lattice coupling \(\beta \to \infty\) — is the key open problem. We argue that the Latent framework provides a natural setting for this program because the grade structure reformulates the CQFT problem as a single analytical question: is \(\rho(\beta) \to \infty\) as \(\beta \to \infty\)?

If the answer is yes, the Osterwalder-Schrader axioms, the Wightman reconstruction, and the existence of a mass gap all follow from the grade decay bound [COND]. The Latent framework thereby reduces the Clay Millennium Yang-Mills problem to a statement about a single real-valued function [CONJ].

We validate the framework on the Schwinger model (QED\(_2\)), the simplest exactly solvable gauge theory with a mass gap. We construct its explicit grade decomposition in Lean 4 (zero sorry), prove it satisfies all six constructive QFT requirements simultaneously, and recover the known mass gap \(m = e/\sqrt{\pi}\) from the grade-2 spectral structure [LEAN]. This is the first gauge theory where the Latent framework is validated against exact results.

Furthermore, we derive an explicit, closed-form formula for the analyticity parameter of U(1) lattice gauge theory: \(\rho(\beta) = \sqrt{\beta^2 + j_1^2}\), where \(j_1 \approx 2.4048\) is the first zero of the Bessel function \(J_0\) [LEAN]. This connects Bessel function zeros to the convergence of the grade expansion — a result we believe is new. Rema