Note that other efforts, including [9, 10, 11, 12, 13], reported a quantitative match in the context of a broad complexity research, but never directly with any standard holographic complexity proposal.

Indeed, gauging $P\to P+2\pi$ is also responsible for selecting $|L=0⟩$ as the true gravitational vacuum state of the theory, since all negative lengths states are removed from the physical Hilbert space [20]. This reduces the Euclidean periodicity ${\beta }_{\mathrm{BH}}=2\pi /\sin (\theta )$ of the black hole background (3) to $\beta ={\beta }_{\mathrm{DSSYK}}=(2\pi -4\theta )/\sin (\theta )$, reproducing the true microscopic temperature of the DSSYK model. In contrast, the Euclidean time periodicity of the solution (3) corresponds to the so called “fake temperature” ${\beta }_{\mathrm{BH}}={\beta }_{\mathrm{fake}}=[2\pi /\sin (\theta )]$ [31].

Without loss of generality, we set the DSSYK coupling $J$ that appears as a multiplicative factor in $E(\theta )$ to $J=1$ for the rest of this Letter. Hence, all times are understood to be measured in units of $J$.

While recent arguments suggest that the identification $|K_n⟩=|n⟩$ holds only for times shorter than those exponential in the entropy [37], we argue that any corrections to this correspondence lie beyond the strict $N\to \infty$ limit of the DSSYK model. This perspective aligns with the expectation that late-time deviations arise from higher-genus contributions to the gravitational path integral of sine dilaton gravity. Since such contributions are not considered in our analysis, the identification of the Krylov and chord bases remains robust within the scope of our Letter.

We point out that the $q\to 1$ limit in our framework is different from the triple-scaling limit $(q\to 1,\theta \ll 1)$, since we are keeping $\theta$ fixed. We regard it as a semiclassical limit because, after rescaling the dilaton in the sine dilaton action (2) as $2|\log q|\mathrm{\Phi }\to \mathrm{\Phi }$ which is our convention in this Letter, $|\log q|$ stands on the same footing as $G_{\mathrm{N}}$.

The answer (14) is not the full quantum mechanical length in JT gravity, which can be obtained by acting as in (10) on the exact JT gravity two-point function [33, 34] and would be given by the triple-scaling limit of (11). Equation (14) then only follows after the $G_{\mathrm{N}}\to 0$ limit of the exact JT answer [42].