Is the string to black hole transition smooth for Schwarzschild black holes?
In plain words
If a black hole shrinks until its horizon is as small as a fundamental string, it should turn into a single highly excited string, and the two entropy formulas roughly agree there. Whether this crossover is smooth, with an exact match of states, is not established.
Precise statement
In weakly coupled string theory in 4D asymptotically flat space, follow a Schwarzschild black hole as the string coupling $g_s$ decreases at fixed entropy until $r_s \sim l_s$. Determine whether the black hole connects continuously to a self-gravitating highly excited string state (Horowitz-Polchinski solution), with the entropy matched including $O(1)$ factors. An answer is a controlled computation of the free energy across $r_s \sim l_s$, or a proof of a phase transition.
What would settle it
A computation of the thermodynamics across the correspondence point in $D = 4$ with controlled corrections in $g_s$ and $\ell_s/r_s$.
Status in the literature
Unverified note
Chen, Maldacena and Witten (2021) mapped where self-gravitating string solutions exist and how they connect to black holes; exact matching across the transition is open as of 2026.