Symbolic Computations of the Two-Colored Diagrams for Central Configurations of the Planar N-vortex Problem

The uploaded paper develops an extended polynomial system (7), the singular-sequence framework, and 11 matrix rules (Theorems 5.3–5.12) for zw-matrices, and outlines a symbolic algorithm that, for N=5, yields exactly 31 two-colored diagrams split into a list of 9 and a list of 22; the nine are said to be excludable by further arguments to appear elsewhere. All of these points are clearly stated in the paper, with proofs or precise rule statements (e.g., Theorems 5.7–5.12) and an explicit algorithmic pipeline (Section 5.2), culminating in the N=5 outcome (Section 6) . The model’s solution matches the high-level narrative (extended system, 11 rules, algorithm, and the 31 = 9 + 22 split) , but it contains substantive technical errors: (i) it flips the asymptotic dictionary for strokes, asserting “z-stroke ⇔ Wjk ≍ ε−2” instead of the correct “z-stroke ⇔ Zjk ≍ ε−2” (since Zjk = 1/wjk and a z-stroke means wjk ≈ ε2) ; (ii) it misstates the formal content of the Second Triangle, Quadrilateral, and Dumbbells rules, replacing the paper’s trace constraints (e.g., Tr(A) > 3, Tr(A) > 4, Tr(A)+Tr(B) > 8; Theorems 5.10–5.12) with “cannot be isolated in B” assertions that the paper does not make in those theorems ; and (iii) parts (3)–(4) of the model’s “Fully Edged Components” rule do not match Theorem 5.9’s statements derived from Section 4 (Corollary 4.4, Propositions 4.7, 4.10, 4.9) . In short, the paper’s statements and algorithmic claims are internally consistent and supported in-text, while the model’s solution introduces incorrect technical formulations even though its conclusions align broadly with the paper.