[
  {
    "id": "ARR-2026-7993ZVQWNC9E38P2-v1",
    "type": "article",
    "title": "Rank-two obstructions to Lorentzian graph partition functions",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          17
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-7993ZVQWNC9E38P2",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-7993ZVQWNC9E38P2/versions/v1/",
    "note": "Research preprint; ARR-2026-7993ZVQWNC9E38P2 v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "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."
  }
]