Arithmetic dynamics of a discrete Painlevé equation

Using the matrices A0, A1 given in (3.2), the top-left entry of A0 contains a standalone −w. Plugging this into the definition I1 = Tr A(1) − (t + 1) from (3.1) yields I1 = y − x + x/y − t/x − w, so I1 depends on w. This directly contradicts Remark 3.3 (“dependence on w drops out”) and the explicit w-free formulas listed for I1–I4. Furthermore, the proof step claiming “since the eigenvalues of A0 are {0, t}, b0 = Tr A0^r = t^r” is false for the printed A0, whose trace is t − w, not t. These contradictions indicate a mis-specification of A(z) in the paper (most likely the extraneous −w term). With the standard qPI Lax normalization where w appears only in mutually reciprocal off-diagonal entries, the model’s outlined argument correctly yields (1) independence of w, (2) invariance under the qPI step (via conjugation when s^r = 1), and (3) bidegree (2r, 2r). See the paper’s definitions of Ir and A(z) and the claimed independence of w and eigenvalue-based step (3.1)–(3.3) and proof lines (Ir(z) = tr + Ir z^r + z^{2r}; b0 = t^r; b2 = 1) , the Lax pair and B(z), w-update (3.4)–(3.5) , and the w-free explicit integrals I1–I4 printed by the authors which conflict with the supplied A0 (3.2) .