Pythagorean Triples in the Fibonacci Model Set

The paper’s Theorem 4.7(a) states an iff classification in the Fibonacci model set using the standard parametrization x=±2lmn, y=l(m^2−n^2), z=l(m^2+n^2), together with the window condition σ(x),σ(y),σ(z)∈[−1,τ−1). As printed, it omits the ‘or with x and y interchanged’ caveat that the authors explicitly included earlier in their Z[τ]-classification (Theorem 3.3). Because of this omission, 4.7(a) is false as written: for example, (x,y,z)=(τ,0,τ) is a Pythagorean triple in Λ (since σ(τ)=τ′∈[−1,τ−1) by Definition 2.1), but it does not fit the displayed orientation; it does fit the swapped orientation x=l(m^2−n^2), y=±2lmn with l=τ, m=1, n=0. Thus 4.7(a) needs the same ‘swap x and y’ proviso used in Theorem 3.3 to be correct. See Theorem 4.7(a) and Definition 2.1 for the window, and compare with Theorem 3.3 where the swap is present . On the other hand, part (b) of 4.7 and its supporting lemmas (4.2–4.6) correctly show eventual entrance into Λ upon scaling by powers of τ via the contraction by τ′ on the conjugate side . The model rightly flags the missing ‘swap’ in (a) and gives a valid direct proof of (b), but it asserts an overly strong degeneracy claim (“y=0 occurs exactly when 2|z”), which fails in the swapped orientation (the same example (τ,0,τ) has z a unit, not divisible by 2). Hence both the paper (statement-level omission in 4.7(a)) and the model (degenerate-case misclassification) are incomplete.