On an existence problem of periodic points in intervals whose images cover themselves

The paper states and claims Theorem 1.3 (existence of a period ≤ 5 point for any covering system of five intervals), but provides only three prototype case analyses and then asserts, without enumerating or proving the remaining cases, that all possibilities can be handled similarly (we claim here without proof that all the cases can be checked..., thus Theorem 1.3 holds), leaving the main proof incomplete . The reduction to discrete problems (Theorems 1.5 and 1.8) is argued, but the central conjecture driving that reduction remains unproven and so cannot fill the gap . The model’s solution builds a digraph from componentwise intersections and then constructs pullback sets along a directed cycle; however, its nonemptiness and surjectivity claims implicitly require Ms+1 ⊆ f(Ms), which do not follow from mere nonempty intersections f(Ci) ∩ Cj ≠ ∅, so the construction can collapse and the proof fails.