Information-Capacity Saturation and Black-Hole Entropy: Constitutive Response and Higher-Derivative Diagnostics

We propose an information-capacity formulation of black-hole entropy in which the Bekenstein--Hawking entropy is interpreted as the saturation value of a coarse-grained information capacity associated with a horizon cross-section. For a stationary black hole in a two-derivative Einstein theory in $D$ spacetime dimensions, we define\begin{equation*}\cI_{\max}=\frac{A}{4G_D\ln 2},\qquad \nuocc=\frac{\Scg}{\SBH}=\frac{4G_D\Scg}{A},\end{equation*}where $\Scg$ denotes the coarse-grained gravitational entropy and $\SBH=A/(4G_D)$. We propose that the maximally coarse-grained equilibrium branch satisfies $\nuocc=1$. Remaining on this branch gives\begin{equation*}\dd A=4G_D\,\dd\Scg,\end{equation*}which we interpret as a constitutive response relation rather than as an independent microscopic derivation of the area law. We test the organization against established black-hole results. At leading order, the five-dimensional D1--D5--P black hole satisfies\begin{equation*}\frac{A_H}{4G_5}=2\pi\sqrt{N_1N_5N_p},\end{equation*}matching the leading microscopic statistical entropy in the appropriate regime. We then generalize the construction to higher-derivative gravity by replacing the Bekenstein--Hawking entropy with the Wald entropy $\SW$. As a literature benchmark, the subleading statistical entropy calculation of Castro and Murthy for the five-dimensional D1--D5--P system agrees with the Wald entropy of the corresponding $R^2$-corrected supergravity theory. We reformulate this established matching as a saturation diagnostic without claiming a new microscopic derivation of the correction. As an explicit effective-theory calculation, we analyze a five-dimensional Einstein--Gauss--Bonnet black hole. We find that the ratio of Wald entropy to the Einstein area entropy is\begin{equation*}\frac{\SW}{A_H/(4G_5)}=1+\frac{12\alpha_{\rm GB}}{r_h^2},\end{equation*}while the corresponding differential entropy--area response is\begin{equation*}\frac{\dd\SW}{\dd(A_H/4G_5)}=1+\frac{4\alpha_{\rm GB}}{r_h^2}.\end{equation*}The Wald entropy also satisfies the first law $\dd M=T_H\dd\SW$ for the solution considered. These results provide a concrete realization of the proposed information-capacity diagnostic. The main value of the framework is therefore organizational and diagnostic: it separates microscopic, coarse-grained, geometric, higher-derivative, and generalized entropies; introduces dimensionless occupancy variables; and provides mismatch parameters that can be tested independently. No microscopic derivation of universal information saturation is claimed. The proposal is instead presented as a constitutive hypothesis whose validity must be tested across independent gravitational and quantum-gravitational settings.