Scale Dependent Memory in Cosmological Backreactions, From Local Viscosity to Global Information Drag

This work extends $\Lambda$CDM to a non--Markovian cosmological backreactionmodel by introducing a causal memory kernel that links the present expansion toa weighted history of nonlinear structure. In the Infinite TransformationPrinciple (ITP) framework, a negative, long--horizon kernel can fit late--timeexpansion and growth data while remaining compatible with early--timeinferences, and shifts the CMB--inferred Hubble rate by a few per cent. Separately, direct measurements in IllustrisTNG show that virialisation and bulk flows generate short--horizon, viscosity--like kernels on nonlinear coarse--graining scales. This paper unifies these results in a scale--dependent memory picture. A multiscale Volterra closure is formulated in which the backreaction term$\delta H^{2}(t;L)$ at coarse--graining scale $L$ is sourced by a structuralvariable $\Sigma(t;L)$ through a kernel $K_{L}(\Delta t)$. Under a mildseparability assumption for the structural source across scales,$\Sigma(t;L)\approx S(L)\Sigma_{\rm bg}(t)$, the background backreaction can bewritten as a convolution with an effective kernel$\Keff(\Delta t)=\int \dd\ln L\,W(L)S(L)K_{L}(\Delta t)$ that remains a kernelin the usual sense. Even if each $K_L$ is a short--horizon exponential, theirmixture need not be. Using TNG300--1 and TNG50--1, kernels are measured at multiple domain sizes tobuild a ladder of memory parameters $(\tau(L),A(L)\tau(L))$ with bootstrapuncertainties. The results show negative, viscosity--like kernels at allnonlinear scales, with $|A\tau|$ increasing towards smaller $L$ and$\tau(L)\lesssim 0.05$--$0.1\,{\rm Gyr}$. An SDSS DR8 domain analysis reproducesthe same sign and scale trend in the real Universe on$L\sim 60$--$240\,{\rm Mpc}/h$ scales. At the horizon, a Planck 2018 ITP fitfixes a long--memory kernel whose integrated drag$|A_{\rm hor}\tau_{\rm hor}|\simeq 0.03$ accounts for the few--percent reductionin $H_{0}$ relative to flat $\Lambda$CDM. Taken together, these measurements define a ``cosmic memory ladder'' frommicro-- to horizon scales. Simple power--law fits show that the effective memorytime grows roughly as $\tau(L)\propto L^{2/3}$ once the horizon point isincluded, while the integrated drag $|A\tau|$ becomes a slowly varying,scale--dependent coupling. A controlled kernel--mixture demonstration illustrateshow fitting a broad mixture of short--horizon kernels with a single exponentialcan inflate the inferred effective horizon. In this sense, the long--range``information drag'' in non--Markovian cosmology can be read as the infraredlimit of the same virial friction that couples structure and expansion insidethe cosmic web.