Joint profile likelihood ﹩{\rm{\Delta }}{\chi }_{{\rm{prof}}}^{2}﹩ in the (f_⋆,0, f_esc) plane, with nuisance parameters profiled at each grid point. The green dotted line marks the degeneracy ridge. The UVLF shape breaks the degeneracy, truncating the allowed region at ε ≈ 2.5% and excluding the BK23 crisis zone (purple). Seven published solutions are overlaid; only those within the 2σ contour are consistent with the data.