{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R445",
  "content_hash": null,
  "slug": "lee7-claim-homotopy-equivalent-torus",
  "type": "claim",
  "title": "The complex is homotopy equivalent to the 2-torus",
  "summary": "A published torus-grid theorem applies with n = 7 and k = 2, so the answer to the candidate question is yes.",
  "relevance": "For Homotopy type of a Lee-metric Rips complex on the 7 by 7 torus, record lee7-claim-homotopy-equivalent-torus (“The complex is homotopy equivalent to the 2-torus”) records a bound, answer, status fact, or structural consequence. The record states: A published torus-grid theorem applies with n = 7 and k = 2, so the answer to the candidate question is yes.",
  "relevance_source": "recorded",
  "body": "The candidate complex \\(Y\\) is exactly the closed Vietoris-Rips complex \\(\\operatorname{VR}(T_{7,7};2)\\) in the notation of Adams, Adetowubo, Barriga-Acosta, Feng, and Sterling. Their metric on \\(T_{n,n}\\) is the sum of the two circular coordinate distances, and a finite set is a simplex when its diameter is at most the scale. This agrees with the stated Lee metric and clique convention.\n\nTheorem 5.8 of the paper proves\n\\[\n\\operatorname{VR}(T_{n,n};k)\\simeq T^2\n\\qquad\\text{for }k\\geq2\\text{ and }n>3k.\n\\]\nHere \\(k=2\\) and \\(n=7>6=3k\\). Therefore\n\\[\n\\boxed{Y\\simeq T^2}.\n\\]\n\nThe proof classifies the facets in the small-scale range and realizes their complex as the nerve of a good cover of \\(\\mathbb R^2/(7\\mathbb Z)^2\\) by projected \\(l^1\\)-balls. Every nonempty finite intersection in the cover is contractible, so the nerve theorem gives the homotopy equivalence. A simplicial collapse is unnecessary because the published nerve equivalence settles the stated question.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the closed Vietoris-Rips complex of the 7 by 7 torus grid with the l1 circular metric at threshold 2",
    "bounds": {
      "side_length": {
        "min": 7,
        "max": 7
      },
      "lee_threshold": {
        "min": 2,
        "max": 2
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/s00009-025-02945-9",
      "locator": "Henry Adams, Adenike Yeside Adetowubo, Hector Barriga-Acosta, Ziqin Feng, and John Sterling, Vietoris-Rips Complexes of Torus Grids, Mediterranean Journal of Mathematics 22, article 173 (2025), Sections 2.1 and 2.3 for the metric and closed Rips convention, Theorem 5.8 on page 14; arXiv:2502.07134v2"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/s00009-025-02945-9",
    "locator": "Henry Adams, Adenike Yeside Adetowubo, Hector Barriga-Acosta, Ziqin Feng, and John Sterling, Vietoris-Rips Complexes of Torus Grids, Mediterranean Journal of Mathematics 22, article 173 (2025), Sections 2.1 and 2.3 for the metric and closed Rips convention, Theorem 5.8 on page 14; arXiv:2502.07134v2"
  },
  "relations": [
    {
      "slug": "R444",
      "title": "The exact parameter pair is covered by a 2025 torus-grid theorem",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R446",
      "title": "The integral homology is Z, Z squared, Z, 0, 0",
      "object_type": "claim",
      "relation": "implies",
      "direction": "outgoing"
    },
    {
      "slug": "lee-rips-torus-7-homotopy",
      "title": "lee rips torus 7 homotopy",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}