Billiard Partitions, Fibonacci Sequences, SIP Classes, and Quivers

The paper derives an explicit double-sum expression for P_B^even (with an overall q^2 factor) and then states Proposition 4.2 identifying it with the reduced quiver-form series PQ for the 2-node quiver with C=[[2,1],[1,1]] after the specialization x1=q^5 x, x2=−a q^3 x; the remark immediately before allows dropping the global q^2. The candidate solution repeats exactly these steps: it expands PQ, performs the same specialization and reindexing, notes sign cancellation, recognizes the q^2–binomial, and explains the global q^2 normalization. Term-by-term the summand matches the paper’s expression, and the only subtlety—the global q^2—matches Remark 4.1. Hence both are correct and essentially the same argument.