{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R161",
  "content_hash": null,
  "slug": "cset11-artifact-boundary-smith-replay",
  "type": "artifact",
  "title": "Exact reduced boundary and Smith-form replay",
  "summary": "Standard-library Python reconstructs the 45 by 45 boundary matrix and certifies the Smith diagonal with two exact Bareiss determinants.",
  "relevance": "For Largest first-homology torsion from forty-five triangles on eleven vertices, record cset11-artifact-boundary-smith-replay (“Exact reduced boundary and Smith-form replay”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python reconstructs the 45 by 45 boundary matrix and certifies the Smith diagonal with two exact Bareiss determinants.",
  "relevance_source": "recorded",
  "body": "The program parses the face codes, orients each face increasingly, and constructs the reduced boundary matrix in lexicographic edge order. Its Bareiss routine uses integer division at every elimination step. It checks the full determinant 74 and the unit 44 by 44 minor.\n\nThese two determinants suffice for the claimed Smith form. The full determinant is the product of all invariant factors. The unit minor forces the 44th determinantal divisor to equal 1, so the first 44 invariant factors are 1. The final invariant factor is 74.\n\nThe program also binds the face sequence and all matrix entries to separate SHA-256 digests. Its canonical report has digest `bdce1e59cd40f418ee11dc99ca39eacbb63d0dc35029921ca098c2ad40ce620e`, and the standard output has digest `544c1f2666ee94e3df31aa6a10a652ffdb3cc83832b6c273cf295943fa60f013`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "all 2025 entries of the reduced boundary matrix, its determinant, and the specified 44 by 44 Smith minor",
    "bounds": {
      "matrix_rows": {
        "min": 45,
        "max": 45
      },
      "matrix_columns": {
        "min": 45,
        "max": 45
      },
      "matrix_entries": {
        "min": 2025,
        "max": 2025
      },
      "determinants_replayed": {
        "min": 2,
        "max": 2
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_computation",
    "entrypoint": "join source_lines with newline and run with python3",
    "runtime": "CPython 3, standard library only",
    "citation": {
      "url": "https://arxiv.org/abs/1707.09271",
      "locator": "Self-contained CPython standard-library computation produced and replayed on 2026-07-25"
    },
    "inline_source": [
      "from hashlib import sha256",
      "from itertools import combinations",
      "from json import dumps",
      "N=11",
      "CODES='359 038 235 037 68A 57A 248 016 569 058 13A 568 129 127 024 025 236 09A 45A 046 159 147 679 28A 168 46A 189 479 148 238 36A 27A 49A 067 37A 345 789 15A 02A 134 578 267 249 039 01A'.split()",
      "def parse(code):",
      "    face=tuple(sorted(10 if char=='A' else int(char) for char in code))",
      "    assert len(face)==3 and len(set(face))==3 and 0<=face[0]<face[1]<face[2]<N",
      "    return face",
      "faces=list(map(parse,CODES))",
      "assert len(faces)==len(set(faces))==45",
      "edges=list(combinations(range(1,N),2))",
      "edge_index={edge:i for i,edge in enumerate(edges)}",
      "matrix=[[0]*45 for _ in range(45)]",
      "for column,(a,b,c) in enumerate(faces):",
      "    for edge,value in (((b,c),1),((a,c),-1),((a,b),1)):",
      "        if edge[0]>0:",
      "            matrix[edge_index[edge]][column]=value",
      "assert sum(value!=0 for row in matrix for value in row)==111",
      "def bareiss(source):",
      "    a=[row[:] for row in source]",
      "    size=len(a)",
      "    assert all(len(row)==size for row in a)",
      "    if size==0:",
      "        return 1",
      "    previous=1",
      "    sign=1",
      "    for k in range(size-1):",
      "        pivot_row=next((row for row in range(k,size) if a[row][k]),None)",
      "        if pivot_row is None:",
      "            return 0",
      "        if pivot_row!=k:",
      "            a[k],a[pivot_row]=a[pivot_row],a[k]",
      "            sign=-sign",
      "        pivot=a[k][k]",
      "        for i in range(k+1,size):",
      "            for j in range(k+1,size):",
      "                numerator=a[i][j]*pivot-a[i][k]*a[k][j]",
      "                assert numerator%previous==0",
      "                a[i][j]=numerator//previous",
      "        previous=pivot",
      "        for i in range(k+1,size):",
      "            a[i][k]=0",
      "    return sign*a[-1][-1]",
      "determinant=bareiss(matrix)",
      "minor=[[matrix[i][j] for j in range(45) if j!=1] for i in range(45) if i!=0]",
      "minor_determinant=bareiss(minor)",
      "assert determinant==74 and minor_determinant==-1",
      "smith_diagonal=[1]*44+[74]",
      "faces_raw='\\n'.join(CODES)+'\\n'",
      "matrix_raw='\\n'.join(','.join(map(str,row)) for row in matrix)+'\\n'",
      "faces_sha=sha256(faces_raw.encode()).hexdigest()",
      "matrix_sha=sha256(matrix_raw.encode()).hexdigest()",
      "assert faces_sha=='cdd0175fca03a927e3ae5b11da108dc2ca61d045f15a4c3442ecfa9bbc928f85'",
      "assert matrix_sha=='a3b4e556bd16ddd806f8199a4aad171acb521102d12e3a27e3f65beb8b991c15'",
      "report={'boundary_shape':[45,45],'determinant':determinant,'faces_sha256':faces_sha,'kalai_upper_bound':3**18,'matrix_sha256':matrix_sha,'nonzero_entries':111,'smith_diagonal':smith_diagonal,'torsion_order':abs(determinant),'unit_minor_determinant':minor_determinant,'unit_minor_removed_column':1,'unit_minor_removed_row':0}",
      "payload=dumps(report,sort_keys=True,separators=(',',':'))",
      "assert sha256(payload.encode()).hexdigest()=='bdce1e59cd40f418ee11dc99ca39eacbb63d0dc35029921ca098c2ad40ce620e'",
      "print(payload)"
    ],
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1707.09271",
    "locator": "Self-contained CPython standard-library computation produced and replayed on 2026-07-25"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R164",
      "title": "A 45-face complex has H1 isomorphic to Z/74Z",
      "object_type": "claim",
      "relation": "verifies",
      "direction": "outgoing"
    },
    {
      "slug": "complete-skeleton-eleven-torsion",
      "title": "complete skeleton eleven torsion",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}