{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R471",
  "content_hash": null,
  "slug": "lshac-artifact-cell-and-grid-enumeration",
  "type": "artifact",
  "title": "Exact cell construction and finite-grid enumeration",
  "summary": "Standard-library Python verifies the continuous 3/16 construction and exhausts five finite cyclic half-set problems.",
  "relevance": "For Sharp long-shift autocorrelation for half-measure circle sets, record lshac-artifact-cell-and-grid-enumeration (“Exact cell construction and finite-grid enumeration”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python verifies the continuous 3/16 construction and exhausts five finite cyclic half-set problems.",
  "relevance_source": "recorded",
  "body": "The program represents a cyclic half-set by a bit mask. For every grid shift \\(j\\), the intersection count is the population count of the mask intersected with its cyclic rotation by \\(j\\). It verifies the five construction counts and uses the linear interpolation formula to certify the continuous supremum.\n\nIt also enumerates every half-subset for grid orders 4, 8, 12, 16, and 20. The exact discrete optima are respectively\n\\[\n1/4,\\quad2/8,\\quad3/12,\\quad3/16,\\quad4/20.\n\\]\nThese discrete lower bounds apply only to unions of cells on the named grid. The enumeration gives no rounding theorem for arbitrary measurable sets.\n\nFor each order, the artifact sorts all optimizing index tuples and hashes their newline-delimited representation. The eight-line standard output has SHA-256 digest `2dceff6f0feed5733f8c45f87f2a24cc126a203d79abc1373a797ff3667fe868`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "the explicit 16-cell set and all half-subsets of cyclic grids of orders 4, 8, 12, 16, and 20",
    "bounds": {
      "minimum_grid_order": {
        "min": 4,
        "max": 4
      },
      "maximum_grid_order": {
        "min": 20,
        "max": 20
      }
    },
    "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/0711.0572",
      "locator": "Inline CPython standard-library computation executed on 2026-07-24"
    },
    "outputs": "construction_cells=0,1,2,3,4,5,7,10\nconstruction_counts_j4_to_j8=3,3,3,3,2\ncontinuous_supremum=3/16\ngrid_n=4 optimum=1/4 optimizers=4 sha256=083908b5f44954e1d15e5efb00137fbd020be0e250046ddbdc38756c76dbef32\ngrid_n=8 optimum=2/8 optimizers=56 sha256=bf85b2d20dadf471fcc222d8fe01d21d66e89de30faa33b78b0010c658878d21\ngrid_n=12 optimum=3/12 optimizers=288 sha256=f52ccc75e428943c24568dc4f2fa697e467a882989219c19122bc81a684f7cd0\ngrid_n=16 optimum=3/16 optimizers=64 sha256=1784633b358fd35c298f5f1319566ba9a60b4a66ee77789919840fa275dd7af6\ngrid_n=20 optimum=4/20 optimizers=1120 sha256=1f4d1e917df8fe859a4ee8c50224142ad7cbe9b889bd557dde3cb464d9758c9a\n",
    "inline_source": [
      "from hashlib import sha256",
      "from itertools import combinations",
      "def popcount(x): return bin(x).count('1')",
      "def rotate(mask,j,n):",
      "    full=(1<<n)-1",
      "    return ((mask<<j)|(mask>>(n-j)))&full",
      "def intersection_count(mask,j,n):",
      "    return popcount(mask&rotate(mask,j,n))",
      "construction=(0,1,2,3,4,5,7,10)",
      "mask=sum(1<<i for i in construction)",
      "counts=[intersection_count(mask,j,16) for j in range(4,9)]",
      "assert counts==[3,3,3,3,2]",
      "print('construction_cells='+','.join(map(str,construction)))",
      "print('construction_counts_j4_to_j8='+','.join(map(str,counts)))",
      "print('continuous_supremum=3/16')",
      "expected={",
      "4:(1,4,'083908b5f44954e1d15e5efb00137fbd020be0e250046ddbdc38756c76dbef32'),",
      "8:(2,56,'bf85b2d20dadf471fcc222d8fe01d21d66e89de30faa33b78b0010c658878d21'),",
      "12:(3,288,'f52ccc75e428943c24568dc4f2fa697e467a882989219c19122bc81a684f7cd0'),",
      "16:(3,64,'1784633b358fd35c298f5f1319566ba9a60b4a66ee77789919840fa275dd7af6'),",
      "20:(4,1120,'1f4d1e917df8fe859a4ee8c50224142ad7cbe9b889bd557dde3cb464d9758c9a')}",
      "for n in (4,8,12,16,20):",
      "    best=n+1",
      "    optimizers=[]",
      "    for chosen in combinations(range(n),n//2):",
      "        candidate=sum(1<<i for i in chosen)",
      "        score=max(intersection_count(candidate,j,n) for j in range(n//4,n//2+1))",
      "        if score<best:",
      "            best=score; optimizers=[chosen]",
      "        elif score==best:",
      "            optimizers.append(chosen)",
      "    raw=''.join(','.join(map(str,item))+'\\n' for item in optimizers)",
      "    digest=sha256(raw.encode()).hexdigest()",
      "    assert (best,len(optimizers),digest)==expected[n]",
      "    print(f'grid_n={n} optimum={best}/{n} optimizers={len(optimizers)} sha256={digest}')"
    ],
    "missing": [
      "command"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/0711.0572",
    "locator": "Inline CPython standard-library computation executed on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R475",
      "title": "An explicit half-set has long-shift supremum 3/16",
      "object_type": "claim",
      "relation": "verifies",
      "direction": "outgoing"
    },
    {
      "slug": "R473",
      "title": "The constant lies between 1/6 and 3/16",
      "object_type": "claim",
      "relation": "tests",
      "direction": "outgoing"
    },
    {
      "slug": "long-shift-autocorrelation-half-set",
      "title": "long shift autocorrelation half set",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}