The Golden Ratio and Hydrodynamics

The paper’s Theorem 4 states that the cusp bifurcation for two equal-strength vortices in a half-plane occurs at W=1/φ for a pair and W=φ for a dipole, and that at the cusp the vortices lie on a common vertical and the cross-ratio CR(z1,z2, z̄2, z̄1) equals φ. The paper proves this by: (a) deriving the vertical alignment at instantaneous stop via the Hamiltonian equations (Lemma 5), (b) solving the ẋ=0 condition on that vertical to get y1/y2 = 2λ + sqrt(4λ^2+1) (Lemma 6), which yields y1/y2 ∈ {φ^3, φ^{-3}} for λ=±1, and (c) evaluating W at that geometry and computing the cross-ratio (with a supporting “balance of actions” observation). The candidate solution follows the same route: it derives the vertical alignment via ẏi ∝ (x1−x2)(1/B^2−1/C^2), imposes ẋ1=0 at x1=x2 to obtain the same quadratic relations in y1/y2, computes CR=4y1y2/(y1−y2)^2=φ, and evaluates W to get W_pair=1/CR=1/φ and W_dip=1+1/CR=φ. The steps, identities, and conclusions match the paper’s derivations and results, with the model providing slightly more explicit algebra (e.g., a direct CR formula). Therefore, both are correct and use substantially the same proof line. Key claims and formulas align with the paper’s statements and calculations for Lemmas 5–6 and Theorem 4.