Upper Bounds on the Sizes of Finite Orbits for Unramified Morphisms

The paper claims full decidability of S-finiteness for unramified endomorphisms by exhibiting an effective bound C(X,R,m) on the size of any finite S-orbit and then asserting that one can decide by computing O_⌈C⌉ and checking whether |O_⌈C⌉| ≤ C (else not S-finite). This last step is logically invalid: an infinite orbit can still satisfy |O_⌈C⌉| ≤ C (e.g., two order-2 automorphisms on P^1 generating an infinite group yield |O_n| growing only linearly in n, so at n = ⌈C⌉ the size can be ≤ C). The bound theorems (Theorems 6, 7) and the reduction/mod m lifting arguments look sound and interesting, but the decidability algorithm as written does not follow. A correct decision procedure would need to check stabilization by time C (e.g., O_C = O_{C+1}), which the paper does not state. The model’s assessment that the general problem was not known to be decidable (and only semi-decidable in general) remains correct given the flaw in the paper’s main decidability step.