[
  {
    "id": "ARR-2026-3WVWP20DY09HFASA-v1",
    "type": "article",
    "title": "Rank plus Weyl sparsity is at least p + 1: an operator form of the Chebotarev-Tao uncertainty principle in prime dimension",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          9
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-3WVWP20DY09HFASA",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-3WVWP20DY09HFASA/versions/v1/",
    "note": "Research preprint; ARR-2026-3WVWP20DY09HFASA v1. Not peer reviewed by AIRR.",
    "abstract": "Let $p$ be prime and let $W_{(a,b)}=X^aZ^b$ be the cyclic Weyl operators on $\\mathbb C^p$. We prove that every nonzero linear combination $C$ of $\\ell$ Weyl operators satisfies $\\dim\\ker C\\le\\ell-1$, i.e. $\\operatorname{rank}C+s_W(C)\\ge p+1$, where $s_W$ counts the nonzero Weyl coefficients. The collinear case is Tao's uncertainty principle on $\\mathbb Z/p$ (Chebotarev's theorem on Fourier minors). The general case follows from the reduction-modulo-$(1-\\omega)$ mechanism of Frenkel and Tao after two further observations: Weyl combinations stay integral under the non-unimodular Clifford conjugations $F'Q_s$, which realise all $p+1$ line directions of $\\mathbb F_p^2$, and, in the nonvacuous sparsity range, a pigeonhole over those directions produces a nonzero fibre sum. The bound is sharp for every $\\ell\\le p$ and fails in every composite dimension. As a self-contained corollary, for one use of the $p^2$ Weyl channels the minimal perfect list length over probes of Schmidt rank $r$ is exactly $p-r+1$. An exact certificate refutes a natural \"line conjecture\"."
  }
]