[
  {
    "id": "ARR-2026-1KTM07SEEV968BCB-v1",
    "type": "article",
    "title": "Sparse cyclic Weyl operators in prime-power dimension: an operator extension of the scalar uncertainty law through sparsity $2p-1$",
    "author": [
      {
        "literal": "Lluis Eriksson"
      }
    ],
    "issued": {
      "date-parts": [
        [
          2026,
          9,
          14
        ]
      ]
    },
    "publisher": "AIRR.SCIENCE",
    "archive": "AIRR.SCIENCE",
    "archive_location": "ARR-2026-1KTM07SEEV968BCB",
    "version": "v1",
    "genre": "Research preprint",
    "URL": "https://airr.science/papers/ARR-2026-1KTM07SEEV968BCB/versions/v1/",
    "note": "Research preprint; ARR-2026-1KTM07SEEV968BCB v1. Not peer reviewed by AIRR.",
    "abstract": "Let $d=p^k$ and let $W_{(a,b)}=X^aZ^b$ be the cyclic Weyl operators on $\\mathbb C^d$. Write $w_p(r)$ for the base-$p$ digit weight $\\prod_t(r_t+1)$ and $\\Omega_k(m)=\\max\\{r<p^k:w_p(r)\\le m\\}$. We give a valuation proof of the classical Delvaux–Van Barel scalar law: (Z) a polynomial with at most $m$ terms and degree below $d$ annihilates at most $\\Omega_k(m)$ of the $d$-th roots of unity, and this is sharp. Our operator extension is: (B) every nonzero combination $C$ of $\\ell\\le2p-1$ Weyl operators satisfies $\\dim\\ker C\\le\\Omega_k(\\ell)$, i.e. $\\operatorname{rank}C\\ge p^k-(\\ell-1)p^{k-1}$ for $\\ell\\le p$ (sharp) and $\\operatorname{rank}C\\ge p^{k-1}$ for $p\\le\\ell\\le2p-1$. The commutative ingredient is the Massey–Costello–Justesen weight-retaining property; the operator ingredient is a Radon-type lemma showing that first-order obstructions have weight at least $2p$. For composite $d$ the maximal nullity is $d/p_{\\min}$ at sparsity two and, conditionally on a published trinomial rank rule, $\\max(2d/r_3,\\,d/2\\,[d\\text{ even}])$ at sparsity three. The conjecture $\\dim\\ker C\\le\\Omega_k(s_W(C))$ for all sparsities is proved for $d\\in\\{4,8,9\\}$ and for torsion cosets; its first open case is $d=16$, $\\ell=6$."
  }
]