{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R443",
  "content_hash": null,
  "slug": "lee7-artifact-exact-cliques-and-smith",
  "type": "artifact",
  "title": "Exact clique enumeration and Smith-normal-form certificate",
  "summary": "The executable computation constructs every clique, finds all facets, and computes the Smith form of every integral simplicial boundary matrix.",
  "relevance": "For Homotopy type of a Lee-metric Rips complex on the 7 by 7 torus, record lee7-artifact-exact-cliques-and-smith (“Exact clique enumeration and Smith-normal-form certificate”) supplies evidence or a replay used to check the packet. The record states: The executable computation constructs every clique, finds all facets, and computes the Smith form of every integral simplicial boundary matrix.",
  "relevance_source": "recorded",
  "body": "The program labels the 49 vertices by pairs in \\(\\{0,\\ldots,6\\}^2\\). It creates an edge exactly when the sum of circular coordinate distances is at most two, then extends cliques in increasing vertex order until none remain. The resulting face vector is\n\\[\n(f_0,f_1,f_2,f_3,f_4)=(49,294,490,294,49).\n\\]\nThere are 49 maximal tetrahedra and 49 maximal four-simplices.\n\nFor each oriented simplex, the program builds the integral boundary by alternating deletion of one vertex. SymPy computes exact Smith normal forms over \\(\\mathbb Z\\). The boundary shapes are\n\\[\n49\\mathbin{\\times}294,\\quad294\\mathbin{\\times}490,\\quad490\\mathbin{\\times}294,\\quad294\\mathbin{\\times}49,\n\\]\nand their ranks are \\(48,244,245,49\\). Every nonzero Smith entry equals one. The report therefore certifies the Betti numbers and absence of integral torsion stated in the homology claim.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "all simplices and all integral boundary matrices of the 49-vertex Lee-threshold-two clique complex",
    "bounds": {
      "side_length": {
        "min": 7,
        "max": 7
      },
      "lee_threshold": {
        "min": 2,
        "max": 2
      },
      "simplex_dimension": {
        "min": 0,
        "max": 4
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_computation",
    "entrypoint": "join source_lines with newline and run in an environment containing SymPy",
    "runtime": "CPython 3 with SymPy 1.14.0",
    "citation": {
      "url": "https://doi.org/10.1007/s00009-025-02945-9",
      "locator": "Inline Python and SymPy computation executed on 2026-07-24 with SymPy 1.14.0"
    },
    "inline_source": [
      "from collections import Counter",
      "from hashlib import sha256",
      "from itertools import combinations",
      "from json import dumps",
      "from sympy import Matrix, ZZ",
      "from sympy.matrices.normalforms import smith_normal_form",
      "vertices=[(i,j) for i in range(7) for j in range(7)]",
      "def circular(a,b):",
      "    d=abs(a-b); return min(d,7-d)",
      "def adjacent(u,v):",
      "    return u!=v and circular(u[0],v[0])+circular(u[1],v[1])<=2",
      "adj=[[adjacent(u,v) for v in vertices] for u in vertices]",
      "simplices=[[(i,) for i in range(len(vertices))]]",
      "while True:",
      "    nxt=[s+(v,) for s in simplices[-1] for v in range(s[-1]+1,len(vertices)) if all(adj[u][v] for u in s)]",
      "    if not nxt: break",
      "    simplices.append(nxt)",
      "f_vector=[len(s) for s in simplices]",
      "assert f_vector==[49,294,490,294,49]",
      "facet_dims=Counter()",
      "for dim,ss in enumerate(simplices):",
      "    for s in ss:",
      "        if not any(v not in s and all(adj[u][v] for u in s) for v in range(len(vertices))): facet_dims[dim]+=1",
      "assert facet_dims=={3:49,4:49}",
      "boundary_ranks=[]; snf_counts=[]; boundary_shapes=[]",
      "for k in range(1,len(simplices)):",
      "    rows={s:i for i,s in enumerate(simplices[k-1])}",
      "    data=[[0]*len(simplices[k]) for _ in simplices[k-1]]",
      "    for col,s in enumerate(simplices[k]):",
      "        for i in range(k+1): data[rows[s[:i]+s[i+1:]]][col]=(-1)**i",
      "    mat=Matrix(data); form=smith_normal_form(mat,domain=ZZ)",
      "    diagonal=[abs(int(form[i,i])) for i in range(min(form.shape)) if form[i,i]]",
      "    boundary_shapes.append(list(mat.shape)); boundary_ranks.append(len(diagonal)); snf_counts.append(dict(sorted(Counter(diagonal).items())))",
      "assert boundary_ranks==[48,244,245,49]",
      "assert snf_counts==[{1:48},{1:244},{1:245},{1:49}]",
      "betti=[]",
      "for k,c in enumerate(f_vector):",
      "    down=boundary_ranks[k-1] if k else 0",
      "    up=boundary_ranks[k] if k<len(boundary_ranks) else 0",
      "    betti.append(c-down-up)",
      "assert betti==[1,2,1,0,0]",
      "report={'vertices':49,'lee_threshold':2,'f_vector':f_vector,'facet_dimension_counts':dict(sorted(facet_dims.items())),'euler_characteristic':sum((-1)**i*n for i,n in enumerate(f_vector)),'boundary_shapes':boundary_shapes,'boundary_ranks':boundary_ranks,'smith_nonzero_invariant_factor_counts':snf_counts,'integral_homology':[{'degree':i,'free_rank':b,'torsion_invariant_factors':[]} for i,b in enumerate(betti)]}",
      "payload=dumps(report,sort_keys=True,separators=(',',':'))",
      "assert sha256(payload.encode()).hexdigest()=='be0c2ae87c4bac14241b97e6fdf895d733082de3af805e22576edf8ed5ddbde1'",
      "print(payload)"
    ],
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/s00009-025-02945-9",
    "locator": "Inline Python and SymPy computation executed on 2026-07-24 with SymPy 1.14.0"
  },
  "relations": [
    {
      "slug": "R446",
      "title": "The integral homology is Z, Z squared, Z, 0, 0",
      "object_type": "claim",
      "relation": "reproduces",
      "direction": "outgoing"
    },
    {
      "slug": "lee-rips-torus-7-homotopy",
      "title": "lee rips torus 7 homotopy",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}