Validation of the profile likelihood. (a) ﹩{\rm{\Delta }}{\chi }_{{\rm{prof}}}^{2}﹩ along the ridge; the 2σ boundary is crossed at ﹩{\varepsilon }_{{\rm{\max }}}=2.5 \% ﹩. (b) Profiled nuisance parameters: at high f_⋆,0, the optimizer steepens α_lo and compresses σ_UV. (c) UVLF model versus data at baseline (ε = 1.9%). (d) At the crisis threshold (ε = 3.5%, Δχ² = 13.4): the UVLF shape is visibly distorted. The log-normal model gives Δχ² = 11.8 (3.4σ); the duty-cycle model gives Δχ² = 20.2 (4.5σ).