{
  "abstract": "Let $D=2^n$ and $M=3n+9\\binom n2$. We study the right-invariant one-step-cliff metric $d_Q$ on $\\operatorname{PU}(D)$, with unit penalty on Pauli weights one and two and penalty $Q$ on all higher weights. If $M/Q_D\\to0$ and $MQ_D^{3/4}/D^2\\to0$, then, for every fixed $0<x<π/\\sqrt3$, \\[ μ_D\\bigl(B_{Q_D}([I],x\\sqrt{Q_D})\\bigr)\\le e^{-c_xD^2}. \\] Here $μ_D$ is normalized Haar measure. Thus the Haar-typical distance from the identity and the diameter are both asymptotic to $(π/\\sqrt3)\\sqrt{Q_D}$ throughout the window $M\\ll Q_D\\ll D^{8/3}M^{-4/3}$. Choosing $Q_D=κD^{8/3}M^{-4/3}$ with sufficiently small fixed $κ>0$ yields a Haar-typical lower bound of order $D^{4/3}M^{-2/3}$ for the corresponding infinite-cliff Carnot--Carathéodory distance, outside an $e^{-Ω(D^2)}$ exceptional set. The infinite-cliff diameter therefore has exponential lower rate at least $4/3$, disproving Brown's exponent-one conjecture. The same estimate gives a fixed-error no-ancilla two-qubit circuit lower bound of the same order.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.03323",
  "author_total": 1,
  "authors": [
    {
      "name": "Honghuai Fang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6h9t0gb25hh91.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.03323",
  "fulltext_url": null,
  "id": "6h9t0gb25hh91",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.03323",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "quant-ph"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.03323",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6h9t0gb25hh91.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:01Z",
      "native_id": "2609.03323",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.03323"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "Variable-Cliff Nielsen Geometry and an Exponent -4/3 Lower Bound for the Infinite-Cliff Diameter",
  "title_text": "Variable-Cliff Nielsen Geometry and an Exponent -4/3 Lower Bound for the Infinite-Cliff Diameter",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/6h9t0gb25hh91",
  "venue": "arXiv",
  "versions": []
}