Fredholm determinant for q-Painlevé III₃

The paper defines a one–circle Riemann–Hilbert problem and the Widom determinant tau τ(t) (Def. 4.1), proves the Bäcklund ratio g(t) = (s∞/s0) t^σ q^{σ−1/4} τ/τ̃ (Thm. 4.2), and derives the bilinear identity (1−t) τ(qt) τ(q−1 t) = τ(t)^2 + (qt)^{1−2σ}(s0/s∞)^2 q^{−1/2} τ̃(t)^2 together with its companion and the standard renormalized form (Thm. 4.3) . The candidate solution follows the same overall strategy: RH formulation, finite-rank variation of τ under t-shifts, Bäcklund dressing, and recovery of the nonlinear P(A(1)'₇) g-equation. Two minor inaccuracies appear: (i) the paper’s conjugated projector update is rank-1 (not “≤2”) and directly yields a scalar ratio, cf. L0^{-1} Π+ L0 = Π+ + u ⊗ ū and τ(qt)/τ(t) = 1 + … ; and (ii) the renormalization prefactor for T_ren is s0/s∞ (not its inverse) in order to obtain the symmetric √t-Hirota pair . With these adjustments, the candidate’s proof aligns with the paper’s argument and conclusions.