We develop a gauge-covariant stochastic effective field theory for stability and finite-width effects in deep neural systems. The model uses classical commuting fields: a complex matter field, a real Abelian connection field, and a fictitious stochastic depth variable. Using the Martin--Siggia--Rose--Janssen--de~Dominicis formalism, we derive its functional representation and a two-replica linear-response construction defining the maximal Lyapunov exponent and the amplification factor for the edge of chaos. Finite-width effects appear as perturbative corrections to dressed kernels, and the marginality condition remains unchanged at the order considered for fixed kernel geometry. Numerically, finite-width multilayer perceptrons follow the mean-field instability threshold, and a linear stochastic effective sector reproduces the predicted low-frequency spectral deformation.

翻译：我们发展了一种规范协变的随机有效场论，用于研究深度神经系统的稳定性与有限宽度效应。该模型采用经典对易场：一个复物质场、一个实阿贝尔联络场以及一个虚构的随机深度变量。利用Martin-Siggia-Rose-Janssen-de Dominicis形式体系，我们推导了其泛函表示以及定义一个双副本线性响应结构，该结构定义了最大李雅普诺夫指数和混沌边缘的放大因子。有限宽度效应表现为对微扰饰核（dressed kernels）的修正，在固定核几何形状的考虑阶数下，边缘性条件保持不变。数值上，有限宽度多层感知机遵循平均场不稳定阈值，而线性随机有效扇区再现了预测的低频谱形变。