Discrete Covariant Derivative and Path Integral in Neural Network Training: Isomorphism with Lattice Gauge Theories

Lattice quantum chromodynamics (LQCD) is a discrete gauge theory in which the matter field is defined on nodes and the gauge field (connection) on edges. We establish an exact mathematical isomorphism between this structure and convolutional neural networks: lattice nodes correspond to neurons, edges to synaptic connections, and network weights are interpreted as a discrete connection. The signal-propagation equation in the network is a discrete analogue of the covariant derivative. We then argue that dropout training can be viewed as a computable approximation to a functional integral over weights with the action S[θ]=βL(θ)+regularisationS[θ]=βL(θ)+regularisation, and discuss the conceptual status of this analogy. Building on Titov's subject-centred model, we propose a quantitative criterion for minimal subjectivity: positive integrated information Φ>0Φ>0 arising from a closed causal loop (reentry). Code and a formal Lean 4 proof of a simplified probabilistic version are appended.