{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R218",
  "content_hash": null,
  "slug": "dpb101-artifact-eighteen-point-cover-check",
  "type": "artifact",
  "title": "Exact replay of the 18-element product basis",
  "summary": "A dependency-free Python program checks the generator order, all 153 pairs, and both 100-element images.",
  "relevance": "For Minimum distinct-product basis of F_101, record dpb101-artifact-eighteen-point-cover-check (“Exact replay of the 18-element product basis”) supplies evidence or a replay used to check the packet. The record states: A dependency-free Python program checks the generator order, all 153 pairs, and both 100-element images.",
  "relevance_source": "recorded",
  "body": "The program first certifies that 2 has order 100 by checking the two prime divisors of 100. It then computes the powers indexed by the displayed exponent set. Every unordered pair is evaluated once in the exponent group and once in the field. Both images must equal their full target sets.\n\nThe multiplicity distribution of the modular sums is 70 classes represented once, 15 represented twice, 12 represented three times, two represented four times, and one represented nine times. These counts account for all 153 pairs. The canonical output line has SHA-256 digest `90143a42f6b43f9dde66574827f9954b8b90d7b9a43aa5d2d40d14e840c68ff0`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "all unordered pairs of the displayed 18 exponents and their corresponding powers of 2 in F_101",
    "bounds": {
      "field_order": {
        "min": 101,
        "max": 101
      },
      "exponent_count": {
        "min": 18,
        "max": 18
      },
      "unordered_pairs": {
        "min": 153,
        "max": 153
      },
      "target_elements": {
        "min": 100,
        "max": 100
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "complete",
    "kind": "inline_python_computation",
    "command": "python3 check.py",
    "runtime": "Python 3 standard library",
    "citation": {
      "url": "https://cs.uwaterloo.ca/journals/JIS/VOL7/Haanpaa/haanpaa.html",
      "locator": "Inline Python 3 computation executed on 2026-07-24"
    },
    "outputs": "order=100;exponents=0,1,4,10,16,18,23,29,36,50,57,63,68,70,76,82,85,86;residues=1,2,16,14,88,49,53,59,78,100,74,90,52,6,81,33,62,23;pairs=153;sum_image=100;product_image=100;min_mult=1;max_mult=9\nsha256=90143a42f6b43f9dde66574827f9954b8b90d7b9a43aa5d2d40d14e840c68ff0\n",
    "inline_source": "from collections import Counter\nimport hashlib\n\nP = 101\nEXPONENTS = [0, 1, 4, 10, 16, 18, 23, 29, 36, 50, 57, 63, 68, 70, 76, 82, 85, 86]\nEXPECTED_RESIDUES = [1, 2, 16, 14, 88, 49, 53, 59, 78, 100, 74, 90, 52, 6, 81, 33, 62, 23]\nEXPECTED_DIGEST = \"90143a42f6b43f9dde66574827f9954b8b90d7b9a43aa5d2d40d14e840c68ff0\"\n\nassert pow(2, 100, P) == 1\nassert pow(2, 50, P) != 1\nassert pow(2, 20, P) != 1\nassert len(EXPONENTS) == len(set(EXPONENTS)) == 18\nresidues = [pow(2, exponent, P) for exponent in EXPONENTS]\nassert residues == EXPECTED_RESIDUES\nassert len(set(residues)) == 18\nsum_counts = Counter()\nproduct_counts = Counter()\nfor i in range(18):\n    for j in range(i + 1, 18):\n        sum_counts[(EXPONENTS[i] + EXPONENTS[j]) % 100] += 1\n        product_counts[(residues[i] * residues[j]) % P] += 1\nassert set(sum_counts) == set(range(100))\nassert set(product_counts) == set(range(1, P))\nassert sum(sum_counts.values()) == sum(product_counts.values()) == 153\nassert Counter(sum_counts.values()) == Counter({1: 70, 2: 15, 3: 12, 4: 2, 9: 1})\nsummary = (\n    \"order=100;\"\n    \"exponents=\" + \",\".join(map(str, EXPONENTS)) + \";\"\n    \"residues=\" + \",\".join(map(str, residues)) + \";\"\n    \"pairs=153;\"\n    f\"sum_image={len(sum_counts)};product_image={len(product_counts)};\"\n    f\"min_mult={min(sum_counts.values())};max_mult={max(sum_counts.values())}\"\n)\ndigest = hashlib.sha256(summary.encode()).hexdigest()\nassert digest == EXPECTED_DIGEST\nprint(summary)\nprint(\"sha256=\" + digest)\n"
  },
  "formal_statement": null,
  "source": {
    "url": "https://cs.uwaterloo.ca/journals/JIS/VOL7/Haanpaa/haanpaa.html",
    "locator": "Inline Python 3 computation executed on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R220",
      "title": "The minimum lies between 15 and 18",
      "object_type": "claim",
      "relation": "evidences",
      "direction": "outgoing"
    },
    {
      "slug": "distinct-product-basis-f101",
      "title": "distinct product basis f101",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}