{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R736",
  "content_hash": null,
  "slug": "sps101-claim-exact-maximum-nine",
  "type": "claim",
  "title": "The exact maximum is 9",
  "summary": "A nine-element set meets both Sidon conditions, while the multiplicative condition alone excludes size 10.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "The answer is\n\\[\n\\max |A|=9.\n\\]\nA witness is\n\\[\nA=\\{1,11,47,62,67,70,71,84,96\\}.\n\\]\nIts 45 unordered pair sums with repetition are distinct in \\(\\mathbb F_{101}\\), and its 45 corresponding products are also distinct.\n\nFor the upper bound, 2 generates \\(\\mathbb F_{101}^{*}\\), which has order 100. Write every element of a product-Sidon set as \\(2^b\\), with \\(b\\in\\mathbf Z/100\\mathbf Z\\). Distinct pair products are equivalent to distinct pair sums of the exponents. This is also equivalent to requiring all directed differences \\(b_i-b_j\\), for \\(i\\ne j\\), to be distinct. Thus a size-10 product-Sidon set would give a \\((100,10)\\)-modular Golomb ruler.\n\nBuratti and Stinson's complete classification through order 11 states that \\((v,10)\\)-modular Golomb rulers do not exist for \\(92\\leq v\\leq106\\). In particular, there is no \\((100,10)\\)-ruler. The executable artifact independently exhausts the case \\(v=100\\). Since any simultaneous set must satisfy the product condition, its size is at most 9. The displayed set attains that bound.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "subsets of the nonzero elements of F_101 whose unordered pair sums and unordered pair products, with repetition, are separately distinct",
    "bounds": {
      "field_order": {
        "min": 101,
        "max": 101
      },
      "maximum_size": {
        "min": 9,
        "max": 9
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.26493/1855-3974.2374.9ff",
      "locator": "Marco Buratti and Douglas R. Stinson, New results on modular Golomb rulers, optical orthogonal codes and related structures, Ars Mathematica Contemporanea 20 (2021), 1-27, definition of a modular Golomb ruler and Table 1; independently replayed in sps101-artifact-witness-and-size-ten-exhaustion"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.26493/1855-3974.2374.9ff",
    "locator": "Marco Buratti and Douglas R. Stinson, New results on modular Golomb rulers, optical orthogonal codes and related structures, Ars Mathematica Contemporanea 20 (2021), 1-27, definition of a modular Golomb ruler and Table 1; independently replayed in sps101-artifact-witness-and-size-ten-exhaustion"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R734",
      "title": "Exact witness replay and exhaustive ten-mark exclusion",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R735",
      "title": "The multiplicative upper bound is a published modular-ruler result",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "simultaneous-sum-product-sidon-f101",
      "title": "simultaneous sum product sidon f101",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}