Minimal Geodesics under Finite Covers

A minimal closed geodesic need not have a simple projection under a finite Riemannian cover. We give explicit flat double covers and primitive minimal geodesics whose primitive projections have exactly m double points, for every positive integer m. Both manifolds can be chosen orientable in every dimension at least three. These examples disprove the simple-projection clause of Theorem 2.5 in Contreras and Mazzucchelli's Closed geodesics and the first Betti number (DOI 10.1112/plms.70085). The obstruction is that a deck isometry need not preserve the minimizing cohomology class. We also prove a sufficient repair: if that class is pulled back from the base, the projected geodesic is minimal and its primitive traversal is simple. The authors' non-cover perturbation theorem then applies on the base. The general entropy-density question AIM 8.4.1 is not resolved, and the truth of the cited general Corollary 2.6 is not decided. This English preprint is AI-assisted, self-audited and unrefereed; novelty remains undetermined. No independent human review, formal verification or absolute-priority claim is made.