A note on the Turing universality of homogeneous potential wells and geodesible flows

The paper’s construction of a homogeneous potential V (for k ≠ 0,2) matches both the tangential and radial components required to obtain ∇V(Q) = −Y2Q (after arranging W⟨−Y2Q+ΔxQ,Q⟩ = k⟨−Y2Q+ΔxQ,WQ⟩), via an explicit v and V1 on the sphere that enforce Euler’s homogeneous identity; the candidate solution, however, only secures exactness of F·dQ on N and then assumes the crucial radial identity F·ω = −kΦ/r without deriving it, leading to a gap. In addition, the model embeds with fields constant in x (hence into HomWellk), which generally requires extra conditions the model does not verify, contradicting the paper’s analysis.