Arithmetic Hypergraph Reconstruction from GCD and LCM Data

Let H=(V,E) be a finite hypergraph whose vertices carry positive integers, and let each edge record the GCD and LCM of its incident labels. We ask which edge-GCD assignments are realizable and when the paired GCD-LCM data determine the vertex labels. The edge-GCD map is right adjoint to an incident-LCM map on the divisibility lattice. The adjunction gives a recognition criterion and the least element of every nonempty GCD fiber. After applying prime valuations, the full fiber is described by nested superlevel sets subject to level-dependent forbidden-edge conditions. On the complete r-uniform hypergraph, this description becomes truncation at the rth largest valuation. The paired transform is injective precisely when the map S to (E[S],E[V]) is injective; any failure already appears on labels in {1,p}^V for a fixed prime p. For graphs, injectivity is equivalent to every connected component being non-bipartite, and an odd-cycle formula then reconstructs all labels without integer factorization. The sharp labeled threshold for K_n^(r) is n>=2r-1, while unordered GCD and LCM decks on divisor chains have threshold n>=2r-2. Each prescribed GCD fiber also has an Euler-product Dirichlet series. If singleton edges are absent, the fiber has positive natural density, with generic local factor determined by the independence polynomial of H.