Let F be a finite connected simple graph with at least one edge such that deleting any edge leaves its independence number unchanged. For every integer s at least two, we prove that the clique blowup F[K_s] admits a strictly positive rational rank-two antiferromagnetic target on six states whose graph partition polynomial is not Lorentzian. The proof classifies minimum-induced-edge subsets using equality in Turan's theorem, identifies an exact four-profile block of inertia (2,2,0), and compresses all higher profiles by a scaled generalized Vandermonde matrix. This yields obstructions for connected biregular bipartite bases other than K2 and for connected bases with a unique maximum independent set and at least one edge. Exact computations check 994 base graphs, 720 stable blowups and four positive rational six-state Hessian witnesses. Six is an upper bound; optimality, uniform target parameters and the general connected-source characterization remain open.
Working paper. Citable, but not yet screened for the AIRR collection.
Research paperWorking paperARR-2026-7993ZVQWNC9E38P2 · v1
We give rank-two antiferromagnetic obstructions to the Lorentzian property of graph partition functions. A connected noncomplete source of maximum degree Delta has a strictly positive rational obstruction on Delta+1 spins whenever no two maximum-degree vertices have equal closed neighbourhoods. This covers graphs without true twins, unique maximum-degree vertices and connected noncomplete edge-regular graphs. Conditioned-set Hessian kernels also yield a neighbourhood-independence-gap criterion. One fixed nonnegative four-spin target characterizes cliques among connected diamond-free and complete multipartite sources; a fixed five-spin target does so for joins of a nonempty clique with disjoint nonempty cliques. Positive perturbations may depend on the source. Exact certificates cover all 989 connected noncliques of orders three through seven. An explicit ten-vertex graph shows that the small-parameter criterion is incomplete. The connected-source conjecture and completeness of the full hierarchy remain open.
Working paper. Citable, but not yet screened for the AIRR collection.
