{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R377",
  "content_hash": null,
  "slug": "ho668-artifact-replay-mod64",
  "type": "artifact",
  "title": "Exact replay of the 64-modular order-668 matrix",
  "summary": "Standard-library Python reconstructs the published matrix, checks every row pair, and reproduces the paper's autocorrelation exceptions and Gram distribution.",
  "relevance": "For A Hadamard matrix of order 668, record ho668-artifact-replay-mod64 (“Exact replay of the 64-modular order-668 matrix”) supplies evidence or a replay used to check the packet. The record states: Standard-library Python reconstructs the published matrix, checks every row pair, and reproduces the paper's autocorrelation exceptions and Gram distribution.",
  "relevance_source": "recorded",
  "body": "The script expands the two run-length encodings in Fact 3.1, applies the stated half-sign switch, builds four circulant matrices, and assembles the Goethals-Seidel blocks. Each row is packed into a Python integer. XOR population counts then evaluate every row product exactly.\n\nThe replay reproduces all thirteen nonzero summed aperiodic autocorrelations in the paper. It verifies the congruence modulo 64 and finds the published 641 zero and 26 nonzero off-diagonal products in every row. The packed matrix uses 84 little-endian bytes per row, with bit \\(j\\) equal to 1 exactly when column \\(j\\) contains \\(+1\\). The concatenated bytes have SHA-256 digest `b9316f8fb407552f6c1301b027e8cddab64796d543801d0005d043cc61a668a1`.\n\nThe six-line stdout has SHA-256 digest `915e94a3afae5b1b4f29f7c59e5fe47f55a13c9ed2d1532eeb13ecc43bbccb55`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "all entries and all unordered row pairs of Eliahou's 64-modular matrix of order 668",
    "bounds": {
      "order": {
        "min": 668,
        "max": 668
      },
      "row_pairs": {
        "min": 222778,
        "max": 222778
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "runnable",
    "kind": "inline_python_computation",
    "command": "python3 check.py",
    "runtime": "CPython 3 standard library",
    "citation": {
      "url": "https://ajc.maths.uq.edu.au/pdf/93/ajc_v93_p422.pdf",
      "locator": "Eliahou 2025, Fact 3.1 and the Gram-matrix statistics on page 426"
    },
    "inline_source": "from collections import Counter\nfrom hashlib import sha256\n\nN = 167\n\ndef expand(runs):\n    out = []\n    sign = 1\n    for length in runs:\n        out.extend([sign] * length)\n        sign = -sign\n    return out\n\nq = expand([83, 2, 81, 1])\ns_runs = [4]*5 + [2,1,1]*5 + [1,5] + [4]*4 + [2,1,1]*6 + [4]*4 + [3] + [1,2,1]*5 + [3] + [4]*4 + [3] + [1,2,1]*5\ns = expand(s_runs)\nassert len(q) == len(s) == N\n\ndef prime(v):\n    h = (len(v) + 1)//2\n    return v[:h] + [-x for x in v[h:]]\n\ndef circ(v):\n    return [v[-i:] + v[:-i] if i else v[:] for i in range(len(v))]\n\ndef tr(M):\n    return [list(row) for row in zip(*M)]\n\ndef xr(M):\n    return [row[::-1] for row in M]\n\ndef neg(M):\n    return [[-x for x in row] for row in M]\n\nA0, B0 = s, prime(s)\nC0 = [x*y for x,y in zip(s,q)]\nD0 = prime(C0)\nA,B,C,D = map(circ, (A0,B0,C0,D0))\nBR,CR,DR = map(xr, (B,C,D))\nBTR,CTR,DTR = map(lambda M: xr(tr(M)), (B,C,D))\nblock_rows = [\n    (A, neg(BR), neg(CR), neg(DR)),\n    (BR, A, neg(DTR), CTR),\n    (CR, DTR, A, neg(BTR)),\n    (DR, neg(CTR), BTR, A),\n]\nH = []\nfor blocks in block_rows:\n    for i in range(N):\n        H.append(sum((block[i] for block in blocks), []))\norder = len(H)\nassert order == 668 and all(len(row) == order for row in H)\npacked = []\nhash_state = sha256()\nfor row in H:\n    word = sum((x == 1) << j for j,x in enumerate(row))\n    packed.append(word)\n    hash_state.update(word.to_bytes((order + 7)//8, 'little'))\nhist = Counter()\nzeros_per_row = []\nfor i, x in enumerate(packed):\n    zeros = 0\n    for j, y in enumerate(packed):\n        if i == j:\n            continue\n        dot = order - 2*bin(x ^ y).count('1')\n        if dot == 0:\n            zeros += 1\n        else:\n            hist[dot] += 1\n    zeros_per_row.append(zeros)\nassert set(zeros_per_row) == {641}\nassert all(value % 64 == 0 for value in hist)\nexpected = {-512:2, -320:2, -256:2, -192:4, -64:4, 128:6, 256:4, 384:2}\nassert hist == Counter({value: count*order for value,count in expected.items()})\n\ndef autocorr(v,k):\n    return sum(v[i]*v[i+k] for i in range(len(v)-k))\ncoeffs = [sum(autocorr(v,k) for v in (A0,B0,C0,D0)) for k in range(1,N)]\nexceptions = [(i+1,c) for i,c in enumerate(coeffs) if c]\nassert exceptions == [(4,-512),(8,384),(12,-256),(16,128),(26,-64),(30,128),(34,-192),(38,256),(42,-320),(46,256),(50,-192),(54,128),(58,-64)]\nprint('order', order, 'entries_pm1', all(abs(x)==1 for row in H for x in row))\nprint('mod64_gram', all(value % 64 == 0 for value in hist), 'true_hadamard', not hist)\nprint('zero_offdiagonal_per_row', min(zeros_per_row), max(zeros_per_row))\nprint('nonzero_dot_multiplicities_per_row', ' '.join(f'{v}:{expected[v]}' for v in sorted(expected)))\nprint('autocorrelation_exceptions', ' '.join(f'{k}:{v}' for k,v in exceptions))\nprint('matrix_pm1_bits_sha256', hash_state.hexdigest())\n",
    "missing": [
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://ajc.maths.uq.edu.au/pdf/93/ajc_v93_p422.pdf",
    "locator": "Eliahou 2025, Fact 3.1 and the Gram-matrix statistics on page 426"
  },
  "relations": [
    {
      "slug": "R380",
      "title": "The 2025 modular construction fails exact orthogonality",
      "object_type": "claim",
      "relation": "validates",
      "direction": "outgoing"
    },
    {
      "slug": "R378",
      "title": "Construction and citation audit",
      "object_type": "attempt",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "hadamard-order-668",
      "title": "hadamard order 668",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}