Information Saturation, Effective Incompressibility, and Higher-Derivative Black-Hole Entropy

We formulate a conservative information-theoretic framework for black-hole thermodynamics in which the Bekenstein--Hawking entropy is interpreted as a maximum coarse-grained information capacity associated with an effective codimension-two surface. The central postulate is not that every individual string microstate possesses a literal local ``one-bit-per-Planck-cell'' structure, but that a maximally coarse-grained black-hole state saturates the capacity\[I_{\max}(\Sigma)=\frac{A(\Sigma)}{4G_D\ln2}.\]Introducing the dimensionless occupancy\[\nu\equiv\frac{I_{\rm cg}}{I_{\max}}=\frac{4G_D S_{\rm cg}}{A},\]we show that saturation, $\nu=1$, together with quasistatic capacity preservation implies the response law\[dA=4G_D\,dS_{\rm cg}.\]Thus the familiar area--entropy relation is reinterpreted as the constitutive equation of an informationally incompressible coarse-grained state. The distinction between this postulate and the Bekenstein--Hawking area law is made explicit: the framework does not derive the numerical coefficient $1/(4G_D)$ from unitarity alone. We then perform a concrete string-theory consistency test using the five-dimensional D1--D5--P system. With standard compactification conventions, the supergravity horizon area and the D1--D5 CFT Cardy entropy satisfy\[\frac{A_H}{4G_5}=2\pi\sqrt{N_1N_5N_p}=S_{\rm CFT},\]so the proposed occupancy is exactly $\nu=1$ at leading two-derivative order. Differentiating the microscopic entropy with respect to the momentum excitation gives the same area response predicted by the saturation postulate. We emphasize, however, that this is a nontrivial consistency test rather than a microscopic derivation of the postulate. The framework is then embedded in local horizon thermodynamics. Substitution of the constitutive entropy variation into Jacobson's local Rindler argument reproduces the Einstein equation with an undetermined cosmological constant. We explicitly do not claim that D1--D5 fluxes fix the five-dimensional cosmological constant. Finally, we formulate a higher-derivative extension in terms of Wald entropy and perform an explicit subleading D1--D5--P calculation in a controlled compactification studied by Castro and Murthy. Their microscopic statistical entropy and the corresponding higher-derivative macroscopic entropy agree in the same large-charge regime. We translate that agreement into the generalized occupancy $\nu_{\rm W}$ and show explicitly that the proposed saturation condition survives the first controlled correction. The result is therefore best regarded as a testable constitutive proposal at the interface of string theory, black-hole thermodynamics, and emergent gravity.