Chi-Squared Geometry for Robust Finite-Blocklength Information and Dispersion Analysis

We develop a column-wise chi-squared geometry for discrete memoryless channels (DMCs) yielding tight, logarithm-free bounds on mutual information, channel dispersion, and finite-blocklength coding rates without evaluating logarithms of the channel matrix. The key parameter is~$\eta$—the worst-case relative deviation of a transition probability from its output marginal, which is small precisely when the channel is close to the fully noisy channel $t_{ij}=s_j$. We prove three main results: (1) a third-order ratio expansion showing $I(X;Y)\mathrm{/}{\chi }^2(X;Y)\to 1\mathrm{/}2$ as $\eta \to 0$ with an $O(\eta )$ skewness correction; (2) a two-sided dispersion equivalence bounding $V(X;Y)$ above and below by ${\chi }^2(X;Y)$ with explicit constants $c_{\pm }(\eta )\to 1$; and (3) a certified robust design rate $R_{\mathrm{c}\mathrm{e}\mathrm{r}\mathrm{t}}(n,\epsilon )$ with total certification gap $O(\eta )+O(\eta \mathrm{/}\sqrt{n})+O(\log n\mathrm{/}n)$. The certified bounds on $I$ and $V$ require only addition, multiplication, division, and square roots; the final rate also uses $Q^{-1}(\epsilon )$.