{
  "abstract": "Boundary conditions can change the dimensional scale of a sharp functional inequality. We determine this change for inverse-square Hardy inequalities on the product lattices $\\mathbb N^k\\times\\mathbb Z^{N-k}$. Their sharp constants satisfy $C_{N,k}\\sim N^2Ψ(k/N)$ uniformly for $1\\le k\\le N$, where $Ψ(α)\\sim2α$ at zero. The law gives linear growth for a fixed number of positive coordinates, every intermediate power for suitable sublinear choices, and quadratic growth for a positive proportion. We develop a method that connects exact uncertainty inequalities with Hardy inequalities through logarithmic localization, and combines this transfer with a block decomposition to cover all proportions. The transfer holds on weighted graphs with an explicit geometric error and controls the variational problem over arbitrary functions. A mixture of two Mathieu ground energies determines $Ψ$. At the fully positive endpoint, the variational problem defining $κ=Ψ(1)$ has a unique nondegenerate minimum and a unique positive energy--moment minimizing profile. We also prove strict comparison with the continuum coefficient in every sector for $N\\ge9$, the optimal uniform dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02206",
  "author_total": 1,
  "authors": [
    {
      "name": "Carlos Lizama"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/560gs1t2mea9e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02206",
  "fulltext_url": null,
  "id": "560gs1t2mea9e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02206",
  "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.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02206",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/560gs1t2mea9e.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02206",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02206"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Hardy transitions on product lattices",
  "title_text": "Hardy transitions on product lattices",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/560gs1t2mea9e",
  "venue": "arXiv",
  "versions": []
}