Virialisation as Viscosity: Deriving the ITP Cosmological Memory Kernel from TNG300

Non--Markovian extensions of $\Lambda$CDM model the backreaction of nonlinear structure on the background expansion through a memory kernel that relates deviations in the effective expansion rate to a structural source built from small--scale dynamics. In the Infinite Transformation Principle (ITP) framework this is written as a Volterra equation for the deviation $\delta H^2(t)$ of the effective expansion rate from the Friedmann prediction, with a kernel $K_{\rm ITP}$ and a structural source $\Sigma(t)$. In previous work, $K_{\rm ITP}$ was introduced phenomenologically and constrained by fits to $H(z)$ and growth data, leading to the criticism that the kernel was not derived from first principles.This paper takes a first step toward a simulation--based derivation. The TNG300-1 simulation is coarse--grained into a $4\times4\times4$ grid of $\sim 50\,{\rm Mpc}/h$ domains. For each snapshot between $z\simeq 2$ and $z=0$ the analysis constructs a domain--averaged velocity--dispersion source $\Sigma(t)\equiv\langle\sigma_v^2\rangle_D$ and an effective expansion--rate deviation $\delta H^2(t)\equiv H_{\rm eff}^2(t)-H_{\rm FRW}^2(t)$ in $(\mathrm{km\,s^{-1}\,Mpc^{-1}})^2$. The relation between these two time series is then modeled as a causal Volterra convolution with an exponential kernel, \[ \delta H^2(t_n) \simeq \sum_{m\le n} K(t_n-t_m)\,\Sigma(t_m)\,\Delta t_m, \qquad K(\Delta t)=A\,\exp(-\Delta t/\tau). \]The best--fit kernel is negative and short--ranged. Using ten snapshots and 64 domains per snapshot, the analysis finds $\tau \simeq 0.51\,\mathrm{Gyr}$, $A\simeq -2.17\times 10^{-3}$, and a 90\% memory horizon $T_{\rm mem}\simeq 1.17\,\mathrm{Gyr}$, with an integrated drag strength $\int_0^\infty K(\Delta t)\,{\rm d}\Delta t = A\tau \simeq -1.1\times 10^{-3}$. A fit that excludes the final $z=0$ snapshot yields $\tau\simeq 0.37\,\mathrm{Gyr}$, $A\simeq -2.82\times 10^{-3}$ and $T_{\rm mem}\simeq 0.86\,\mathrm{Gyr}$ with a similar drag strength, showing that $A\tau$ is more stable than $A$ or $\tau$ taken individually. The negative amplitude implies that increasing $\Sigma(t)$ drives $\delta H^2(t)