{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R657",
  "content_hash": null,
  "slug": "rm28-artifact-exact-cubic-distance",
  "type": "artifact",
  "title": "Exact quotient and Walsh replay for the distance-88 cubic",
  "summary": "A deterministic standard-library program enumerates all 8,192 quotient representatives, checks the complete score histogram, and verifies a nearest quadratic by direct truth-table weight.",
  "relevance": "For Covering radius of the second-order Reed-Muller code RM(2,8), record rm28-artifact-exact-cubic-distance (“Exact quotient and Walsh replay for the distance-88 cubic”) supplies evidence or a replay used to check the packet. The record states: A deterministic standard-library program enumerates all 8,192 quotient representatives, checks the complete score histogram, and verifies a nearest quadratic by direct truth-table weight.",
  "relevance_source": "recorded",
  "body": "Join source_lines with LF, append a terminal LF, save the result as `rm28_exact.py` in a disposable directory, and run the recorded command. The program derives the eight-dimensional invariance span, enumerates the 13-dimensional quotient, evaluates both affine distances by exact integer Walsh transforms, and asserts the full histogram. It then evaluates the displayed 8-variable witness and nearest quadratic directly. Truth tables use integer inputs 0 through 255, with bit i assigned to variable i. The independent bitset implementation in rm28-artifact-bitset-crosscheck tests the same result through exhaustive affine masks.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "the displayed 8-variable cubic against the complete space of degree-at-most-two corrections",
    "bounds": {
      "variables": {
        "min": 8,
        "max": 8
      },
      "quadratic_coefficients": {
        "min": 37,
        "max": 37
      },
      "quotient_representatives": {
        "min": 8192,
        "max": 8192
      },
      "truth_table_rows": {
        "min": 256,
        "max": 256
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "complete",
    "kind": "inline_python_exact_computation",
    "command": "python3 rm28_exact.py",
    "entrypoint": "Join source_lines with LF, append one terminal LF, and save as rm28_exact.py",
    "runtime": "CPython 3.9.6 standard library, macOS 26.2 arm64",
    "citation": {
      "locator": "Self-contained CPython standard-library program authored and executed on 2026-07-28"
    },
    "dependencies": [
      {
        "name": "CPython standard library",
        "version": "3.9.6",
        "license": "Python-2.0"
      }
    ],
    "outputs": {
      "source_sha256": "48e6b03098d3d9e8bebff7e0ec036aa88cd0fca2aeee329f1c328c3c4da96fd9",
      "stdout_sha256": "b49025a58c8a15040dae1584753d54daeee353eb0888af83644d263126c1774f",
      "expected_stdout": "span_rank=8\nquotient_dimension=13\nrepresentatives=8192\nhistogram=88:28,92:1016,96:2968,100:3024,104:1092,108:56,112:8\nminimum=88\nnearest_q=ad+bc+bd+bg\ndirect_weight=88\nslice_weights=40,48\nwitness_truth_sha256=47299b7d07c1a0de9de3c88d211b1b39d6d3d45259662df9aa77c9638874cc26\ncorrected_truth_sha256=22c23f99ad7843999487757749e4f4c9b9879151db4d016c5e91597eccccaea6\n",
      "span_rank": 8,
      "quotient_dimension": 13,
      "representatives": 8192,
      "minimum": 88,
      "histogram": {
        "88": 28,
        "92": 1016,
        "96": 2968,
        "100": 3024,
        "104": 1092,
        "108": 56,
        "112": 8
      }
    },
    "runtime_seconds": 3.427693,
    "inline_source": [
      "from collections import Counter",
      "from hashlib import sha256",
      "from itertools import combinations",
      "",
      "PAIRS = list(combinations(range(7), 2))",
      "PAIR_BIT = {pair: 1 << i for i, pair in enumerate(PAIRS)}",
      "G_TERMS = [(0, 2, 6), (0, 3, 5), (1, 3, 6), (2, 3, 4)]",
      "P_TERMS = [(0, 1), (2, 5), (4, 6), (5, 6)]",
      "F_TERMS = [",
      "    (0, 1, 2),",
      "    (0, 3, 6),",
      "    (0, 5, 7),",
      "    (0, 6, 7),",
      "    (1, 3, 7),",
      "    (1, 4, 6),",
      "    (2, 4, 7),",
      "    (3, 4, 5),",
      "]",
      "NEAREST_TERMS = [(0, 3), (1, 2), (1, 3), (1, 6)]",
      "",
      "",
      "def quadratic_mask(terms):",
      "    mask = 0",
      "    for pair in terms:",
      "        mask ^= PAIR_BIT[tuple(sorted(pair))]",
      "    return mask",
      "",
      "",
      "def anf_table(variable_count, terms):",
      "    return [",
      "        sum(all((x >> i) & 1 for i in term) for term in terms) & 1",
      "        for x in range(1 << variable_count)",
      "    ]",
      "",
      "",
      "def quadratic_table(mask):",
      "    terms = [PAIRS[i] for i in range(len(PAIRS)) if (mask >> i) & 1]",
      "    return anf_table(7, terms)",
      "",
      "",
      "def affine_distance(bits):",
      "    walsh = [1 - 2 * bit for bit in bits]",
      "    step = 1",
      "    while step < len(walsh):",
      "        for base in range(0, len(walsh), 2 * step):",
      "            for i in range(base, base + step):",
      "                left, right = walsh[i], walsh[i + step]",
      "                walsh[i], walsh[i + step] = left + right, left - right",
      "        step *= 2",
      "    return (len(bits) - max(map(abs, walsh))) // 2",
      "",
      "",
      "p_mask = quadratic_mask(P_TERMS)",
      "derivatives = []",
      "for variable in range(7):",
      "    derivative_terms = [",
      "        tuple(i for i in term if i != variable)",
      "        for term in G_TERMS",
      "        if variable in term",
      "    ]",
      "    derivatives.append(quadratic_mask(derivative_terms))",
      "",
      "rows = [p_mask, *derivatives]",
      "rank = 0",
      "pivots = []",
      "for column in range(len(PAIRS)):",
      "    hit = next(",
      "        (i for i in range(rank, len(rows)) if (rows[i] >> column) & 1),",
      "        None,",
      "    )",
      "    if hit is None:",
      "        continue",
      "    rows[rank], rows[hit] = rows[hit], rows[rank]",
      "    for i in range(len(rows)):",
      "        if i != rank and ((rows[i] >> column) & 1):",
      "            rows[i] ^= rows[rank]",
      "    pivots.append(column)",
      "    rank += 1",
      "",
      "free = [i for i in range(len(PAIRS)) if i not in pivots]",
      "assert rank == 8 and len(free) == 13",
      "",
      "g = anf_table(7, G_TERMS)",
      "p = anf_table(7, P_TERMS)",
      "histogram = Counter()",
      "for selector in range(1 << len(free)):",
      "    mask = sum(1 << column for j, column in enumerate(free) if (selector >> j) & 1)",
      "    h = quadratic_table(mask)",
      "    distance = affine_distance([u ^ v for u, v in zip(g, h)])",
      "    distance += affine_distance([u ^ v ^ w for u, v, w in zip(g, p, h)])",
      "    histogram[distance] += 1",
      "",
      "expected = [(88, 28), (92, 1016), (96, 2968), (100, 3024)]",
      "expected += [(104, 1092), (108, 56), (112, 8)]",
      "assert sorted(histogram.items()) == expected",
      "",
      "f = anf_table(8, F_TERMS)",
      "q = anf_table(8, NEAREST_TERMS)",
      "corrected = [u ^ v for u, v in zip(f, q)]",
      "assert sum(corrected) == 88",
      "slice_weights = [sum(corrected[0::2]), sum(corrected[1::2])]",
      "assert slice_weights == [40, 48]",
      "",
      "output = (",
      "    \"span_rank=8\\n\"",
      "    \"quotient_dimension=13\\n\"",
      "    \"representatives=8192\\n\"",
      "    \"histogram=88:28,92:1016,96:2968,100:3024,104:1092,108:56,112:8\\n\"",
      "    \"minimum=88\\n\"",
      "    \"nearest_q=ad+bc+bd+bg\\n\"",
      "    \"direct_weight=88\\n\"",
      "    \"slice_weights=40,48\\n\"",
      "    f\"witness_truth_sha256={sha256(bytes(f)).hexdigest()}\\n\"",
      "    f\"corrected_truth_sha256={sha256(bytes(corrected)).hexdigest()}\\n\"",
      ")",
      "print(output, end=\"\")"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": null,
    "locator": "Self-contained CPython standard-library program authored and executed on 2026-07-28"
  },
  "relations": [
    {
      "slug": "R663",
      "title": "An eight-term cubic has exact second-order nonlinearity 88",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "R656",
      "title": "Independent bitset cross-check of the cubic distance",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "reed-muller-rm2-8-covering-radius",
      "title": "reed muller rm2 8 covering radius",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}