{
  "abstract": "The plunge region records the spectral transition of a time-frequency localization operator. Let $A_0,B_0\\subset\\mathbb{R}$ be bounded measurable sets of positive measure with finite boundaries, and let $S_{cA_0,B_0}=P_{cA_0}Q_{B_0}P_{cA_0}$. We prove that, for every $c>0$ and $0<\\varepsilon<1/2$, $Λ_\\varepsilon\\le C(A_0,B_0)\\widetilde L[1+\\ln_+(ca/\\widetilde L)]$, where $\\widetilde L=\\ln(1/[\\varepsilon(1-\\varepsilon)])$, and $\\ln_+x=\\max(0,\\ln x)$. If $M$ and $K$ count the interval components and $a$ and $b$ are their maximal lengths, one may take $C(A_0,B_0)=63MK(1+b)$. The estimate is uniform in $c$ and $\\varepsilon$ and becomes $O(\\widetilde L)$ when $\\widetilde L\\ge ca$. On the range $c\\ge2$ and $α^{-c}<\\varepsilon<1/2$ covered by the parallelepiped theorem of Kulikov and Dam Larsen (2026), this recovers its $d=1$ order; their complementary very-small-threshold result is sharper. We give an independent direct proof with explicit geometry dependence and a single formulation for all $c$ and $\\varepsilon$. It acts on the off-diagonal factor of the localization operator: an exact one-dimensional oscillation factorization reduces each far-field piece to two modulated copies of the scale-free Hankel kernel $1/[2π(s+r)]$, Bernstein-ellipse approximation gives uniform singular-value decay, and a Taylor-rank estimate controls the boundary layer. Choosing this width at the spectral depth absorbs finer scales and leaves only $\\ln_+(ca/\\widetilde L)$ dyadic scales, yielding the self-improving logarithm. It also identifies tangential phase dependence as an obstruction to this factorization in nonproduct higher-dimensional geometry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.23016",
  "author_total": 1,
  "authors": [
    {
      "name": "Ahmadreza Azimifard"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/266690bdm9kps.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.23016",
  "fulltext_url": null,
  "id": "266690bdm9kps",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.23016",
  "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.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2604|Mathematical Analysis and Transform Methods"
    ]
  ],
  "openalex": "W7171500294",
  "pdf_url": "https://arxiv.org/pdf/2607.23016",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/266690bdm9kps.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.23016",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.23016"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Scale-free Hankel factorization and explicit one-dimensional plunge bounds for time-frequency localization operators",
  "title_text": "Scale-free Hankel factorization and explicit one-dimensional plunge bounds for time-frequency localization operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/266690bdm9kps",
  "venue": "arXiv",
  "versions": []
}