MONOTONICITY AND CRITICAL POINTS OF THE PERIOD FUNCTION FOR POTENTIAL SYSTEM

The paper states and proves: (i) a balance/involution criterion for monotonicity, bounding, and exactness of critical periods in terms of B_sigma(δ); and (ii) degree-dependent bounds for polynomial g, including the odd case and the 'all zeros real' case (Theorems 1.1–1.2). Its core analytic identity for T'(h) is given as an Abel-type transform after a change of variables (formulas (4.5)–(4.6)), from which a variation-diminishing argument is derived to bound zeros of T' by zeros of B_sigma(δ) and to get exactness under B_sigma(G/g^2) conditions. These claims are explicitly stated and supported in the text (Theorem 1.1 and its proof; see statements and derivation around formulas (4.5)–(4.6)). The candidate model reproduces the qualitative shape of these results and tools (involution σ, Abel transform, variation diminishing), but makes two substantive errors: (1) it writes a derivative formula for T'(h) missing the prefactor/weight that appears in the paper (the paper proves sqrt(2h) T'(h) equals an Abel transform of B_sigma(δ) after the change u = G(x), not T'(h) itself; cf. (4.6)), and (2) it incorrectly concludes strict monotonicity (hence zero critical periods) when g has only real zeros, while the paper only proves 'at most one' and exhibits concrete examples with exactly one critical period in that setting. The paper’s statements and examples are coherent; the model’s stronger monotonicity claim is false. See Theorem 1.1 (monotonicity/upper bound/exactness) and Theorem 1.2 (odd polynomials, all zeros real), plus the derivations in Section 4 and examples in Section 5 (e.g., Example 5.3). Citations: Theorem 1.1 and its setup (), derivative identity (4.5) and its ‘Abelized’ version (4.6) (, ), Theorem 1.2 statements (, ), and the one-critical-period examples when all zeros of g are real ().