Spin inference with the Black Hole Explorer. I. Fisher information matrix forecast

General relativity predicts the presence of a thin, bright ring of light superimposed on the image of a black hole (the ``photon ring''), whose geometry is sensitive to the spin of the black hole. Detecting this photon ring in observations of nuclear supermassive black holes would yield insight into gravity in the strong-field regime as well as their growth histories. The Black Hole Explorer (BHEX) is a mission being developed to detect the photon ring for the first time, raising the question of how precisely we can constrain spin from its observations. We use the method of Fisher information matrices (FIMs) along with a simple dual-cone semi-analytic emission model to forecast BHEX spin posterior widths for the primary science targets (M87* and Sgr A*) in a variety of configurations. We find that BHEX can constrain the dimensionless spin of its targets to a precision of σ a* 0.1 after 30 orbits, even in the presence of significant systematic errors. We validate our results via numerical checks for stability and robustness, as well as synthetic data cross-validation to assess handling of covariances and prior information in the FIM against a full posterior exploration.