Properties of Poincaré Half-Maps for Planar Linear Systems via an Integral Characterization

The paper proves sign(y0 + P(y0)) = −sign(T) for y0 ∈ I \ {0} (and also at y0 = 0 under the stated conditions) by introducing the symmetric integral g(u) = ∫_{−u}^{u} (−y/W(y)) dy, noting g′(u) = −2aTu^2/(W(u)W(−u)), and using the integral characterization PV ∫_{P(y0)}^{y0} (−y/W(y)) dy = cT with the correct constants c (0 for a>0, π( D/√(4D−T^2) )^{-1} for a=0, and 2π( D/√(4D−T^2) )^{-1} for a<0). This yields sign(cT − g(u)) = sign(T) and then sign(y0 + P(y0)) = −sign(T) via a decomposition of the integral at −y0. The argument is explicit for a = 0 using P(y0) = −e^{πT/√(4D−T^2)} y0, and it handles a ≠ 0 uniformly via monotonicity of g and the finite limit of g(u) when 4D − T^2 > 0. See equations (4), (7), (13), (17), and Proposition 5 in the paper . The candidate solution mirrors the symmetric-integral reduction and derivative computation but leaves the decisive a < 0 case at the level of an outline, appealing to a “branch selection”/winding-number explanation for the 2π factor without carrying out a rigorous comparison with cT. In contrast, the paper’s proof cleanly avoids branch ambiguities by working directly with g and the integral characterization. Therefore, the paper is correct while the model’s solution is incomplete in the a < 0 regime.