On the periodic decompositions of multidimensional configurations

The paper proves both targets cleanly: (A) a k-periodic decomposition criterion from periodizers with support avoiding every (k−1)-subspace (Theorem 8), and (B) a sparse-product decomposition into periodic fibers (Theorem 12). Key steps include converting a suitable periodizer to an annihilator with the same support constraint (Lemma 7) and then extracting non-parallel difference factors via Theorem 2, followed by a controlled decomposition (Lemma 6) . For the sparse case, the authors handle n=1,2 and then proceed by induction using orbit-closure limits, avoiding any hard “inversion” of operators on fiber-sums (Lemmas 15–16, then Theorem 12) . By contrast, the model’s Part (A) is explicitly incomplete (it lacks the extraction step that the paper supplies via Lemma 7 + Theorem 2), and Part (B) relies on an unjustified invertibility/lifting argument on sums of v-fibers, which the paper circumvents with a different, rigorous method.