The analytically continued hyperbolic matrix-model conjecture

Let $W_n(t;p=4,q)$ be the disk partition function of the matrix model

$$S=N\operatorname{tr}\left(\frac12X^2-\frac1q(\widetilde A X)^q\right),$$

with $\operatorname{tr}\widetilde A^k=Nt\delta_{k,4}$ and loop operator

$$\varphi(X)=\sum_{s=0}^{q-2}(\widetilde A X)^s\widetilde A.$$

Hyperbolic matrix-model conjecture. The disk partition function admits an analytic continuation in $q$ near $q=4$ such that, with

$$n\to\infty,\quad \ell_0^2=\frac{\pi L^2}{2}(q-4)\to0^+,\quad t\to1,\quad n\ell_0^2=\beta_{\mathrm q},\quad \Lambda=\frac{16\pi}{\ell_0^2}(1-t)$$

fixed, it yields the UV-complete disk partition function of pure JT gravity with curvature $R=-2/L^2$. The conjecture proposes an analytic-continuation route from the hyperbolic lattice matrix model to the UV-complete theory, beyond the effective Schwarzian limit.