arXiv2026
The Black Hole-String Correspondence identifies the microstates of a minimal black hole with those of a highly excited string. In decompactification limits, where the light states are Kaluza--Klein modes, its proposed counterpart is the Black Hole-Tower Correspondence at the species scale. We extend it in three directions. First, we expand the free thermodynamic analysis to include towers with string-oscillators and Kaluza--Klein modes, and clarify whether the ensemble connects to a $d$-dimensional black hole or a wrapped black string. Second, we include gravitational backreaction in the pure Kaluza--Klein tower transition. With no winding-tachyon analogue, the interpolating configuration is the one-loop Kaluza--Klein gas, which reorganizes into higher-dimensional radiation and yields self-gravitating solutions whose instability points to the black string and whose thermodynamics match it at the correspondence point, $S\sim N_{\rm sp}$, where the size approaches the species length. This shows that the matching is robust under self-interactions. Third, we ask whether the two phases are continuously connected, revisiting and extending the worldsheet and topological analyses of the Black Hole-String case to include also Kaluza--Klein towers. Linear sigma model families interpolate smoothly in the heterotic string, while in type II the transition is obstructed. The cobordism classes of the two saddles agree in every structure we check, but their brane-charge lattices, given by bordism groups, differ, which is the source of the type II obstruction. This obstruction is thus perturbative, showing that the transition can only proceed through the non-perturbative, charge-violating processes of the kind predicted by the Cobordism Conjecture.