Platonic solids and symmetric solutions of the N-vortex problem on the sphere

The paper’s Theorem 3.5 establishes exactly the claims audited: (i) ρ(Ko,Fo)(S^2) is an embedded, flow-invariant 2–sphere; (ii) restriction of the full flow to this sphere is symplectically conjugate to the reduced Hamiltonian system on (S^2, m ωS^2); (iii) K1-invariance/equivariance when K ≤ K1 ≤ N(K) and F is K1-invariant; (iv) the explicit reduced Hamiltonian h(K,F); and (v) vanishing center of vorticity. The candidate solution reproduces these items using the same core mechanism: ρ^*Ω = m ωS^2 and h = H∘ρ imply X_H ∘ ρ = ρ_* X_h, yielding invariance and conjugacy; the decomposition of H into within-orbit, cross, and fixed terms matches equations (3.7)–(3.8) in the paper; the K1-argument is identical; and J ≡ 0 follows from ∑_{g∈K} g = 0 for K ∈ {Dn, T, O, I}. Minor differences are stylistic (the paper frames invariance via discrete reduction/twisted subgroups, while the candidate deduces it from conjugacy) and the candidate omits the paper’s careful domain caveat (smoothness away from F[K] and collision regularization). Overall, both are correct and essentially the same proof scaffold, with the paper providing additional structural context and technical lemmas (e.g., Lemmas 3.8–3.10). Citations: theorem statement and setup ; definitions of H, Ω, and equations of motion ; symplectic pullback and conjugacy details ; decomposition and explicit h via (3.6)–(3.8) ; K1-invariance argument ; normalizer remark ; center-of-vorticity proof and J definition .