A General Theory of Neural Fields: Width, Attention, Covariance Processes, Renormalization, Gauge Quotients, and Quantum Reconstruction

Abstract

We develop a mathematical framework for neural field laws that separates width, head count, sequence length, depth, training, coarse-graining, parameter quotients, and quantum reconstruction. Under a declared conditional Gaussian representation, connected field correlators are determined by cumulants of the random covariance. Fixed-head attention can consequently remain non-Gaussian at infinite channel width, whereas independent head aggregation suppresses higher cumulants. A grading fan organizes distinct asymptotic paths, and a stationary covariance cocycle identifies the rate of initialization-memory decay. Training is treated separately through controlled Fokker--Planck evolution and Hamilton--Jacobi duality. Explicit coarse maps make RG claims testable. We give transport and composition bounds, exact Gaussian decimation, an exactly closed non-Gaussian Gamma family with computable scaling exponents, and a conditioned spectral estimate for approximate conjugacy. A constructive obstruction shows that zero one-step training risk need not control recursive coarse-graining; discarded-state memory and intermediate-scale support explain two sources of failed closure. Reflection positivity remains a separate requirement. We establish preservation results and negative witnesses, and exhibit a reflection-positive covariance mixture that fails clustering. Neural random fields therefore require additional hypotheses for positive-metric quantum reconstruction. Reproducible calculations include 32 fixed attention kernels and a supervised study of 24 trained models (12 LSTMs and 12 transformers) across three input tasks and three seeds. Intermediate-scale training reduces median eight-token Rademacher averaging error from 0.0665 to 0.0256 for the LSTM and from 0.2664 to 0.00493 for the transformer. All 180 sampled scale records satisfy the applicable trajectory-error bound. These results test recovery of a prescribed map and a compositional failure mechanism; they do not establish spontaneous universality or an isomorphism between generic neural architectures and RG flows.