{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R666",
  "content_hash": null,
  "slug": "rnfc9-artifact-six-vertex-boundary-regression",
  "type": "artifact",
  "title": "Independent labeled boundary-rank regression at six vertices",
  "summary": "A direct Python sweep reproduces all five beta bins and the candidate's positive count of 12,442.",
  "relevance": "This supplies a computation, dataset, or executable check used to assess the target problem.",
  "relevance_source": "derived",
  "body": "This second implementation visits all 32,768 labeled six-vertex graphs directly. It constructs the graph components, creates one integer bit-vector for every triangle boundary, and computes rank by a Python dictionary of XOR pivots. It finds\n\\[\n(\\#\\beta_1=0,\\ldots,\\#\\beta_1=4)\n=(20{,}326,10{,}317,2{,}010,105,10).\n\\]\nThe 12,442 positive cases agree with the candidate's independent six-vertex computation. The canonical report has SHA-256 `4d1011305015f6d7599ee8709932cce455bbd4458ab6b1daefa7fee098f42a63`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "all 2^15 labeled simple graphs on six vertices, evaluated by an independent standard-library implementation of the same chain-rank identity",
    "bounds": {
      "vertices": {
        "min": 6,
        "max": 6
      },
      "labeled_graphs": {
        "min": 32768,
        "max": 32768
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_labeled_enumeration",
    "entrypoint": "Join source_lines with LF characters and execute",
    "runtime": "Python 3 standard library",
    "citation": {
      "url": "https://arxiv.org/abs/math/0605536",
      "locator": "Independent Python 3 standard-library exhaustive sweep executed on 2026-07-25"
    },
    "outputs": "n=6 histogram=0:20326,1:10317,2:2010,3:105,4:10 positive=12442 total=32768\nreport_sha256=4d1011305015f6d7599ee8709932cce455bbd4458ab6b1daefa7fee098f42a63\n",
    "inline_source": [
      "from collections import Counter",
      "from hashlib import sha256",
      "from itertools import combinations",
      "",
      "def beta(n, mask):",
      "    edges=list(combinations(range(n),2)); ix={e:i for i,e in enumerate(edges)}",
      "    p=list(range(n))",
      "    def root(x):",
      "        while p[x]!=x:",
      "            p[x]=p[p[x]]; x=p[x]",
      "        return x",
      "    for i,(u,v) in enumerate(edges):",
      "        if mask>>i&1:",
      "            a,b=root(u),root(v)",
      "            if a!=b: p[a]=b",
      "    c=sum(root(i)==i for i in range(n)); basis={}; rank=0",
      "    for a,b,d in combinations(range(n),3):",
      "        ids=[ix[(a,b)],ix[(a,d)],ix[(b,d)]]",
      "        if all(mask>>i&1 for i in ids):",
      "            x=sum(1<<i for i in ids)",
      "            while x:",
      "                q=x.bit_length()-1",
      "                if q in basis: x^=basis[q]",
      "                else: basis[q]=x; rank+=1; break",
      "    return bin(mask).count('1')-n+c-rank",
      "",
      "h=Counter(beta(6,m) for m in range(1<<15))",
      "report='n=6 histogram='+','.join(f'{k}:{h[k]}' for k in sorted(h))+' positive='+str(sum(v for k,v in h.items() if k))+' total='+str(sum(h.values()))",
      "print(report)",
      "print('report_sha256='+sha256((report+'\\n').encode()).hexdigest())"
    ],
    "missing": [
      "command"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/0605536",
    "locator": "Independent Python 3 standard-library exhaustive sweep executed on 2026-07-25"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R665",
      "title": "Exact nauty orbit and triangle-boundary certificate",
      "object_type": "artifact",
      "relation": "cross_checks",
      "direction": "outgoing"
    },
    {
      "slug": "random-nine-flag-complex-beta-one",
      "title": "random nine flag complex beta one",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}