On the Structure of Periodic Eigenvalues of the Vectorial p-Laplacian

The paper rigorously proves Theorem 1.1 via a scaling reduction to an integrable planar system and the functions Tp(μ), Sp(μ), then uses a careful number-theoretic selection to obtain infinitely many integer‑independent sequences Λm = {(2n̟m)p} (see Theorem 1.1 and its proof outline; Sp,Tp construction; Lemma 4.1; limits as μ→1) . The candidate’s proof sketches a different approach using an apsidal‑angle argument, but makes critical errors: (i) it asserts a 0 lower endpoint for the apsidal increment, contradicting the paper’s limit Sp(μ)→1/2 as μ↓0 (thus Δθ→π per full cycle) ; and (ii) it claims one can choose a subsequence of minimal periods Pm accumulating to 2π to guarantee non‑integer ratios, whereas the paper’s analysis shows that, for closed orbits built from rationals ℓ/m, the minimal periods scale like m·(2πTp(μ)) and in fact ̟m→∞ in the constructed sequences . These gaps invalidate the model’s argument for the integer‑independence property.