Painlevé determinateness theorem

For a first-order ordinary differential equation rational in the dependent variable, with coefficients meromorphic on the chosen independent-variable surface, analytic continuation along an arc ending outside the fixed exceptional set has a definite limit in the Riemann sphere. Its movable singularities are poles or algebraic branch points. To see the limiting assertion, the cluster set of a continuous lifted arc is connected and compact. Any cluster value where the vector field is holomorphic permits uniform local continuation by the Picard-Lindelöf theorem. Otherwise the cluster set lies in finitely many coefficient poles, so it is a singleton. At a pole of the vector field, the inverse equation is holomorphic; its solution has a finite-order critical point, giving a convergent root expansion. The reciprocal dependent variable handles infinity. Points where both reduced numerator and denominator vanish belong to the fixed exceptional set and are excluded from this argument.