In finite-width deep neural networks, the empirical kernel $G$ evolves stochastically across layers. We develop a collective kernel effective field theory (EFT) for pre-activation ResNets based on a $G$-only closure hierarchy and diagnose its finite validity window. Exploiting the exact conditional Gaussianity of residual increments, we derive an exact stochastic recursion for $G$. Applying Gaussian approximations systematically yields a continuous-depth ODE system for the mean kernel $K_0$, the kernel covariance $V_4$, and the $1/n$ mean correction $K_{1,\mathrm{EFT}}$, which emerges diagrammatically as a one-loop tadpole correction. Numerically, $K_0$ remains accurate at all depths. However, the $V_4$ equation residual accumulates to an $O(1)$ error at finite time, primarily driven by approximation errors in the $G$-only transport term. Furthermore, $K_{1,\mathrm{EFT}}$ fails due to the breakdown of the source closure, which exhibits a systematic mismatch even at initialization. These findings highlight the limitations of $G$-only state-space reduction and suggest extending the state space to incorporate the sigma-kernel.

翻译：在有限宽度深度神经网络中，经验核 $G$ 在各层间随机演化。我们基于 $G$ 独有闭合层级发展了一种适用于预激活残差网络的集体核有效场论（EFT），并诊断了其有限的有效性窗口。利用残差增量具有精确条件高斯性这一性质，我们推导出 $G$ 的精确随机递归方程。系统性地应用高斯近似，可得到关于平均核 $K_0$、核协方差 $V_4$ 以及 $1/n$ 均值修正 $K_{1,\mathrm{EFT}}$ 的连续深度常微分方程组，其中 $K_{1,\mathrm{EFT}}$ 在图形化上表现为单圈图修正。数值实验表明，$K_0$ 在所有深度上保持准确。然而，$V_4$ 方程中的残差在有限时间内累积至 $O(1)$ 量级的误差，这主要由 $G$ 独有输运项的近似误差主导。此外，$K_{1,\mathrm{EFT}}$ 因源闭合性失效而失败——即使在初始化阶段，该闭合性也表现出系统性失配。这些发现揭示了 $G$ 独有状态空间约简的局限性，并提出需将状态空间扩展至包含西格玛核。