Horizon Microstructure Thermodynamics in AdS Black Holes: Smarr-Consistent Excitation Enthalpy

In this work we formulate a horizon-microstructure description of four-dimensional AdS black holes in which a horizon of area $A$ is resolved into $\mathcal{N}=A\mathrm{/}{a}_p$ microscopic sites and $N$ occupied horizon sites. The central result is that the combinatorics of this partially occupied horizon sector yields the entropy directly: in the finite-filling regime the leading term is proportional to the area, and the maximal-entropy filling reproduces the Bekenstein–Hawking law with $a_p=4\ln 2{\mathrm{\ell }}_p^2$. Subleading corrections include a subtractive logarithmic term and an inverse-area expansion. We then show that this partially occupied regime admits a thermodynamic justification from an extended first law with chemical potential $\mu$, a Smarr-consistent excitation enthalpy $\delta M(A,P,N)$, and an AdS control parameter $u=PS$. In this interpretation, the combinatorics provides the dominant horizon entropy, while the thermodynamic sector supplies a dressing that selects the equilibrium filling and assigns a finite excitation cost to departures from a reference partially occupied configuration.