Prospects for submillimeter VLBI of black hole photon rings with a stratospheric balloon-borne telescope

Submillimeter very long baseline interferometry (VLBI) achieves higher angular resolution and probes more optically thin emission near supermassive black holes, with less interstellar scattering toward Sgr A$^*$. Above 345 GHz, however, the atmosphere confines ground-based VLBI to a few high, dry sites. Using 2012-2022 weather, we forecast what a 2.5 m telescope floating at 35-40 km, a successor to the 22 GHz Balloon-borne VLBI Experiment (BVEX) pathfinder, would add to ground-VLBI arrays at 345-690 GHz. The balloon adds little baseline length but reaches higher frequencies, where a fixed baseline is longer in wavelengths. On an ALMA-balloon baseline, Sgr A$^*$ at 460 GHz is at the detection threshold, with ten or more 20 s detections on about half of the simulated April days. These detections lie beyond the polarimetric photon ring transition, where the $n=1$ ring dominates the polarized visibility amplitudes. M87$^*$ at 460 GHz would need frequency phase transfer from a 230 GHz reference to be detected. With phase transfer, most Sgr A$^*$ detections beyond the transition at 460 GHz are on balloon baselines, whose thermal noise meets the linear-polarization threshold within one 300 s scan. At 690 GHz such detections require phase transfer that stays coherent at three times the reference frequency. In Fisher forecasts, a 460 GHz ring-geometry constraint requires balloons but is weaker than at 345 GHz, where balloons add little improvement. High-resolution 460 GHz VLBI science would require a 32 GHz bandwidth backend, submillimeter receivers, and simultaneous 230 GHz reference recording at every station.