Monotonicity of Entropy for Unimodal Real Quadratic Rational Maps

The paper proves there are no bone-loops in the unimodal region and deduces monotonicity of real entropy by deploying positive transversality in a family normalized by critical values, then deriving a contradiction from the sign of a directional derivative at two successive intersections on a hypothesized bone-loop; see Theorem 1.1 and the argument around equations (3.2)–(3.6) . The candidate’s solution follows the same overall structure (normalization by critical values, positive transversality, density-of-hyperbolicity to locate a PCF point, then a second critical-relation curve), but replaces the paper’s local directional-derivative contradiction with a global algebraic-intersection-number argument. That argument is essentially sound if one explicitly notes the second curve is a closed 1-cycle in the one-point compactification (S^2), but this closure/compactness point is not stated; the paper’s method avoids that subtlety. Hence both are correct and closely related, with the model using a slightly different (but standard) topological contradiction.