A simple geometric construction of an ODE with undecidable blow-ups

The paper proves undecidability of finite-time blow-up by embedding a discrete torus TM model into a mapping torus flow, then into Euclidean space, and finally using a spherical/radial compactification so that reaching a halting region corresponds to hitting the boundary and hence blowing up; this argument is coherent and complete at the sketch level as presented (Theorem 1.1, torus model, geometric embedding, Poincaré radial compactification) . The candidate solution also achieves an equivalence by a different construction: a robust 2D ODE simulator on S^2 coupled to a blow-up coordinate z′=(z^2+1) gated by a smooth detector; it is correct given a forward-invariant open halting region (as claimed from Graça–Zhong) and standard ODE facts (e.g., blow-up criterion) [Teschl 2012]. Thus both are correct, with distinct methods; we note a minor assumption in the model about the halting region’s forward invariance.