Let (X,dX) and (Y,dY) two nonempty compact Polish spaces, µ 2M +1 (X), 2M +1 (Y) two probability measures on these spaces and c: X Y! R+ a nonnegative and continuous function. As X and Y are compact, r(µ,) is tight, then Prokhorov's theorem applies and the closure of r(µ,) is sequentially compact. Let us now show that r(µ,) is closed. Indeed, Let ( n)n 0 a sequence of r(µ,) converging towards . In addition as ( n)n 0 live in the simplex r, we can also extract a sub-sequence, such that n! 2 r.

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.

Several recent papers considered a related question of a function's sensitivity to its various inputs [5,6,7].

Due to itsiconic nature (i.e., perceptual resemblance to or natural association with the referent), drawings serve as a powerful tool to communicate concepts transcending language barriers (Fay et al., 2014). In fact, we humans started to use drawings to convey messages dating back to 40,000-60,000 years ago (Hoffmann et al., 2018; Hawkins et al., 2019).