[
  {
    "id": "ARR-2026-161TZ0CP488XTTBN-v1",
    "type": "article",
    "title": "Sparse Weyl operators: a uniform binary rank theorem and sharp erasure thresholds",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          10,
          15
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-161TZ0CP488XTTBN",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-161TZ0CP488XTTBN/versions/v1/",
    "note": "Research preprint; ARR-2026-161TZ0CP488XTTBN v1. Not peer reviewed by AIRR. Working paper; not admitted to the AIRR accepted collection.",
    "abstract": "For a complex matrix expanded in the cyclic Weyl basis, we refine the product rank bound using the largest nonzero translate overlap of its support. Combining a fourth-moment estimate, cyclic compression and acyclic block selection, we prove that every nonzero combination of at most seven cyclic Weyl operators in dimension d=2^k has rank at least d/4. For d>=4 this is sharp under every support budget from four through seven, uniformly in k. It completes the conjectured rank envelope in dimension 16 and closes additional gaps in dimensions 27 and 32. Two exact universal erasure budgets follow for arbitrary complex low-rank matrices: seven in a binary family of ranks, and eight for rank at most two in dimension 27; explicit examples fail at the next erasure. A quantitative fourth-moment bound also controls distance from low rank. The proofs use classical scalar uncertainty and graph lemmas. No general prime-power conjecture, density-matrix recovery theorem, efficient recovery algorithm or Erdos problem is claimed solved."
  }
]