{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R220",
  "content_hash": null,
  "slug": "dpb101-claim-certified-interval-15-18",
  "type": "claim",
  "title": "The minimum lies between 15 and 18",
  "summary": "Pair counting gives the lower endpoint, while a new 18-element exponent set covers every class modulo 100.",
  "relevance": "For Minimum distinct-product basis of F_101, record dpb101-claim-certified-interval-15-18 (“The minimum lies between 15 and 18”) records a bound, answer, status fact, or structural consequence. The record states: Pair counting gives the lower endpoint, while a new 18-element exponent set covers every class modulo 100.",
  "relevance_source": "recorded",
  "body": "Write every nonzero field element as \\(2^t\\). The congruences\n\\[\n2^{20}=95,\\qquad 2^{50}=100,\\qquad 2^{100}=1\\pmod {101}\n\\]\nshow that 2 has order 100. A set of field elements is therefore a distinct-product basis exactly when its discrete logarithms form a strict sum cover of \\(\\mathbf Z/100\\mathbf Z\\).\n\nIf the basis has \\(k\\) elements, its distinct unordered pairs produce at most \\(\\binom{k}{2}\\) products. Covering all 100 nonzero elements requires \\(\\binom{k}{2}\\geq100\\), hence \\(k\\geq15\\).\n\nFor the upper bound, take the exponent set\n\\[\nB=\\{0,1,4,10,16,18,23,29,36,50,57,63,68,70,76,82,85,86\\}.\n\\]\nThe 153 sums \\(b+b'\\pmod {100}\\), with \\(b,b'\\in B\\) and \\(b\\ne b'\\), contain every residue class. Equivalently, the field set\n\\[\n\\{1,2,6,14,16,23,33,49,52,53,59,62,74,78,81,88,90,100\\}\n\\]\nhas all 100 nonzero field elements among its products of distinct members. Thus the unknown minimum \\(m\\) satisfies\n\\[\n15\\leq m\\leq18.\n\\]\nThis improves the candidate record's certified upper bound of 19. It leaves sizes 15, 16, and 17 unresolved.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all distinct-product bases in the multiplicative group of F_101, together with the displayed 18-element witness",
    "bounds": {
      "field_order": {
        "min": 101,
        "max": 101
      },
      "group_order": {
        "min": 100,
        "max": 100
      },
      "certified_minimum": {
        "min": 15,
        "max": 18
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://cs.uwaterloo.ca/journals/JIS/VOL7/Haanpaa/haanpaa.html",
      "locator": "Elementary counting and the exact replay in dpb101-artifact-eighteen-point-cover-check"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://cs.uwaterloo.ca/journals/JIS/VOL7/Haanpaa/haanpaa.html",
    "locator": "Elementary counting and the exact replay in dpb101-artifact-eighteen-point-cover-check"
  },
  "models": [],
  "relations": [
    {
      "slug": "R218",
      "title": "Exact replay of the 18-element product basis",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R219",
      "title": "The exact value at order 100 remains open in this audit",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "distinct-product-basis-f101",
      "title": "distinct product basis f101",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}