{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R393",
  "content_hash": null,
  "slug": "hr4-claim-published-n7-rank-lower-bounds",
  "type": "claim",
  "title": "Propagation gives rational rank bounds 3107 and 110 at n=7",
  "summary": "The Adams-Virk propagation theorems, applied to the complete rational n=6 table, give b_7(VR(Q_7;4);Q) at least 3107 and b_15(VR(Q_7;4);Q) at least 110.",
  "relevance": "For Integral torsion in scale-four hypercube Rips complexes, record hr4-claim-published-n7-rank-lower-bounds (“Propagation gives rational rank bounds 3107 and 110 at n=7”) records a bound, answer, status fact, or structural consequence. The record states: The Adams-Virk propagation theorems, applied to the complete rational n=6 table, give b_7(VR(Q_7;4);Q) at least 3107 and b_15(VR(Q_7;4);Q) at least 110.",
  "relevance_source": "recorded",
  "body": "Adams and Virk report the same numerical bounds in Table 2 over the coefficient field used for the \\(n=6\\) computation. Their Theorems 4.1, 6.4, and 6.5 also apply to the later complete rational table.\n\nDegree 7 first appears at cube dimension 6 because the complexes through dimension 5 have homology only in degrees 0 and 15. Theorem 6.4 therefore multiplies the rational rank 239 by\n\\[\n1+2\\binom65=13,\n\\]\ngiving \\(b_7(\\operatorname{VR}(Q_7;4);\\mathbb Q)\\ge3107\\).\n\nIn degree 15, the twelve coordinate copies of \\(Q_5\\) in \\(Q_6\\) give twelve independent cross-polytopal classes. The rational rank 14 at \\(n=6\\) leaves a quotient of rank 2. At \\(n=7\\), Theorem 4.1 supplies 84 cross-polytopal classes and Theorem 6.5 propagates the two additional classes with factor 13. Thus\n\\[\nb_{15}(\\operatorname{VR}(Q_7;4);\\mathbb Q)\\ge84+26=110.\n\\]\nThese are lower bounds. They do not give a complete rational homology table at \\(n=7\\).",
  "status": "supported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "rational homology rank lower bounds in degrees 7 and 15 for VR(Q_7;4)",
    "bounds": {
      "cube_dimension": {
        "min": 7,
        "max": 7
      },
      "scale": {
        "min": 4,
        "max": 4
      },
      "coefficient_characteristic": {
        "min": 0,
        "max": 0
      },
      "degree_7_rank": {
        "min": 3107
      },
      "degree_15_rank": {
        "min": 110
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2309.06222",
      "locator": "Adams and Virk, Theorems 4.1, 6.4, and 6.5 and Table 2; Galetto, Montaño, and Wellner arXiv:2606.20784v1, Appendix C.3"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2309.06222",
    "locator": "Adams and Virk, Theorems 4.1, 6.4, and 6.5 and Table 2; Galetto, Montaño, and Wellner arXiv:2606.20784v1, Appendix C.3"
  },
  "relations": [
    {
      "slug": "R392",
      "title": "Published field computations for VR(Q_6;4)",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R391",
      "title": "The n=7 complex has 209,570,782,049 faces and reduced Euler characteristic -3937",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "hypercube-rips-scale-four-torsion-free",
      "title": "hypercube rips scale four torsion free",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}