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We study subsystem entropies in 2d CFTs for subsystems constituting a finite fraction of the full system. We focus on the extensive contribution, which scales linearly with the subsystem size in the thermodynamic limit. We employ the so-called diagonal approximation to evaluate subsystem entropy for chaotic CFTs in the thermal state (canonical ensemble), the microcanonical ensemble, and in a primary state, matching previously known results. We then proceed to find analytic expressions for the subsystem entropy at leading order in $c$, when the global CFT state is the KdV-generalized Gibbs ensemble or the KdV-microcanonical ensemble. Previous studies of primary eigenstates have shown that, akin to the fixed-area states in AdS/CFT, the corresponding subsystem entanglement spectrum is flat. This behavior is seemingly in sharp contradiction with that of the thermal (microcanonical) state, and thus in apparent contradiction with the subsystem eigenstate thermalization hypothesis (ETH). In this paper, we resolve this issue by comparing the primary state with the KdV-(micro)canonical ensemble. We show that the results are consistent with the KdV-generalized version of the subsystem ETH, in which local properties of quantum eigenstates are governed by their values of conserved KdV charges. Our paper solidifies evidence for the KdV-generalized ETH in 2d CFTs and emphasizes Rényi entropy as a sensitive probe of the reduced-density matrix. 

The eigenstate thermalization hypothesis (ETH) in chaotic two-dimensional CFTs is subtle due to the presence of infinitely many conserved KdV charges. Previous works have shown that primary CFT eigenstates exhibit a flat entanglement spectrum, which is very different from that of the microcanonical ensemble. This appears to contradict conventional ETH, which does not account for KdV charges. In a companion paper \cite{1}, we resolve this discrepancy by analyzing the subsystem entropy of a chaotic CFT in KdV-generalized Gibbs and microcanonical ensembles. In this paper, we perform parallel computations within the framework of AdS/CFT. We focus on the high-density limit, which corresponds to the thermodynamic limit in conformal theories. In this regime, holographic Rényi entropy can be calculated using the so-called *gluing construction*. We specifically study the KdV-generalized microcanonical ensemble where the densities of the first two KdV charges are fixed: $$ \langle Q_1 \rangle = q_1, \quad \langle Q_3 \rangle = q_3 $$ with the condition $$q_3 - q_1^2 \ll q_1^2$$. In this regime, we find that the refined Rényi entropy $$\tilde{S}_n$$ becomes independent of $$n$$ for $$n > n_{\text{cut}}$$, where $$n_{\text{cut}}$$ depends on $$q_1$$ and $$q_3$$. By taking the primary state limit $$q_3 \to q_1^2$$, we recover the flat entanglement spectrum characteristic of fixed-area states, consistent with the behavior of primary states. This result supports the consistency of KdV-generalized ETH in 2d CFTs.