{
  "abstract": "Let $Σ_{1} = \\mathbb{N}^*$ and $Σ_{k+1} = \\overline{\\mathbb{N}^* + Σ_{k}}$ be the cumulative depth filtration of $β\\mathbb{N}$, the analogue for $(\\mathbb{N},+)$ of a chain of closed ideals that Protasov and Protasova studied for discrete groups, where strict descent follows from a theorem of Lutsenko and Protasov. For every $k$ we give an explicit set whose closure meets $Σ_{k}$ but not $Σ_{k+1}$. Fix the doubly exponential sequence $e_{n} = 2^{2^{n}}$, partition it into $k$ subsequences $E_{0}, \\dots, E_{k-1}$ by the residue of the index modulo $k$, and set $A_{k} = E_{0} + \\cdots + E_{k-1}$. We prove that any sum $q_{0} + \\cdots + q_{k-1}$ of free ultrafilters with $E_{t} \\in q_{t}$ lies in $Σ_{k} \\setminus Σ_{k+1}$. The engine is a master lemma, proved by induction on $j$: if a sum $F_{1} + \\cdots + F_{j}$ of subsequences of $\\{e_{n}\\}$ with pairwise disjoint index sets belongs to a free ultrafilter $s$, then $s \\notin Σ_{j+1}$. The proof rests on a single rigidity of the doubly exponential sequence: a fixed difference forces the largest index in any shift-intersection, once it is large, to cancel within its own subsequence, which makes every shift-intersection descend by at least one level. The same witnesses lie in the gaps of the pure filtration.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.38696",
  "author_total": 1,
  "authors": [
    {
      "name": "Carl Aza"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8sn1d2z78k2hf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.38696",
  "fulltext_url": null,
  "id": "8sn1d2z78k2hf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.38696",
  "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.CO"
    ],
    [
      "arxiv",
      "math.GN"
    ],
    [
      "arxiv",
      "math.GR"
    ],
    [
      "arxiv",
      "math.LO"
    ],
    [
      "openalex",
      "2608|Advanced Topology and Set Theory"
    ]
  ],
  "openalex": "W7215101914",
  "pdf_url": "https://arxiv.org/pdf/2609.38696",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8sn1d2z78k2hf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:49Z",
      "native_id": "2609.38696",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.38696"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GR",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GN",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "Explicit Witnesses at Every Gap of the Depth Filtration of $β\\mathbb{N}$",
  "title_text": "Explicit Witnesses at Every Gap of the Depth Filtration of βℕ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:49Z",
  "url": "https://research.draconitic.com/paper/8sn1d2z78k2hf",
  "venue": "arXiv",
  "versions": []
}