Musical tone coloring via bifurcation control of Eulerian n-tuple Hopf singularities

The paper reduces the Eulerian system on each invariant leaf M_{ccc} to the scalar radial equation r1' = r1 f̃(µ,r1) for T^{n+1}-invariant f, then proves: (i) a non-standard pitchfork at µ=0 with stability determined by the sign of the first nonzero even-power coefficient; and (ii) a double saddle-node with transition value µ⋆ for f̃(µ,r1)=α(µ+a r1^{2p}+b r1^{2q}), including the bijection between positive roots and invariant hypertori. These are stated and proved in Lemma 3.3, Corollary 3.5, Theorem 3.6, and Theorem 3.7, with the same reduction (3.10) and the same µ⋆ formula and bifurcation counts . The candidate solution mirrors these steps, derives the same µ⋆ and nondegeneracy condition, and uses the same root–torus correspondence, differing only in presentation and minor explanatory details. Accordingly, both are correct with substantially the same proof.