ODD INDEX OF THE AMENDED POTENTIAL IMPLIES LINEAR INSTABILITY

The paper’s Theorem 1 (odd nullity or odd Morse index of B implies JB is linearly unstable) is stated and proved correctly, with two independent arguments—via spectral flow/relative Morse index and via an elementary invariant-subspace approach . The candidate solution’s Step 1 (odd nullity forces a non-semisimple zero eigenvalue) is correct. However, Step 2 incorrectly asserts that M(B) odd forces det B < 0 and hence spectral instability, overlooking the case ν(B) > 0 where det B = 0. The paper explicitly provides spectrally stable counterexamples with M(B) odd (e.g., B = diag{−2, −1, 1, −1, 0, 0}) . Thus the model’s claim “A cannot be spectrally stable” for odd M(B) is false in general; linear instability in that regime follows from a different mechanism (failure of semisimplicity on appropriate invariant subspaces) as shown in the paper .