On the geometry of a 4-dimensional extension of a q-Painlevé I equation with symmetry type A_1^{(1)}

The paper defines the same birational map φ on (P^1)^4 as the model and proves: (i) φ regularizes to a pseudo-automorphism after 28 blow-ups, (ii) ω = dq1∧dp1/(q1p1) + dq2∧dp2/(q2p2) is preserved on the resolved variety and determines an effective anticanonical divisor D, (iii) the anticanonical linear system on X is two-dimensional and yields conserved quantities, and (iv) a family X_b with an action of W̃(A1^(1))×W̃(A1^(1)) by pseudo-isomorphisms, recovering non-autonomous 4D q-Painlevé-type dynamics from translation elements . However, the model’s blow-up accounting is incorrect: the paper’s explicit formula for −K_X shows exactly eight codimension-3 blow-ups (coefficients 2 on E1, E2, E8, E9, E15, E16, E22, E23) and the remaining twenty are codimension-2 (coefficient 1), not 12 codimension-3 and 16 codimension-2 as claimed by the model . The model also asserts without support that φ̃ is a product of two commuting translations; the paper constructs commuting translation actions in the two A1^(1) factors and derives corresponding non-autonomous systems, but does not state that the autonomous φ̃ itself factors as such a product . Finally, while the model’s A1^(1)⊕A1^(1) Weyl-group description of symmetries matches the paper at the group level, the paper is careful to specify the root/coroot normalization (type (A1⟨α,α∨⟩=1/4 + A1)^(1)), a nuance the model omits .