Non-Perturbative Bounds on Cosmological Backreaction, the Non-Linear Scale, and Gauge-Invariant Mutual Information from the Matter Power Spectrum

We apply the mesoscopic coarse-graining framework of~\cite{OsanoMeso,OsanoExtensivity,OsanoPerturbation} to three problems in Cosmological Perturbation Theory and the backreaction debate. $\mathbf{\text{(i)}}$~A non-perturbative lower bound on the kinematic backreaction $\text{QD}$ in the Buchert equations, derived from the Gibbs–Bogoliubov inequality: $\text{QD}$ cannot be suppressed below its linear-perturbation-theoryvalue, regardless of the degree of non-linearity, provided the system satisfies stability and temperedness. $\mathbf{\text{(ii)}}$~The radius of convergence of the mesoscopic cumulant expansion equals $O({\text{kNL}}^{-1})$, the non-linear scale of the matter power spectrum, providing a KAM-theorem explanation for why standard perturbation theory fails at $k>\text{kNL}$. $\mathbf{\text{(iii)}}$~For a Gaussian matter field, the inter-cell mutual information is exactly $I(i,j)=-\frac{1}{2}\ln (1-r_{ij}^2)$, gauge-invariant at linear order and computable directly from the observed $P(k)$; for $\mathrm{\Lambda }$CDM at $\mathrm{\ell }=50\mathrm{M}\mathrm{p}\mathrm{c}{h}^{-1}$, $I_{\mathrm{N}\mathrm{N}}\approx 0.10$.The total mutual information gives a data-computable measure of the backreaction correction to the FRW free energy. The gauge proof holds at linear order; the KAM identification is exact; the backreaction bound rests on a stated conjecture.