{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R735",
  "content_hash": null,
  "slug": "sps101-attempt-primary-literature-audit",
  "type": "attempt",
  "title": "The multiplicative upper bound is a published modular-ruler result",
  "summary": "The coupled field problem appears absent from the sources checked, while its decisive ten-point obstruction is already known.",
  "relevance": "This records a line of attack on the target problem, including what it establishes and where it stops.",
  "relevance_source": "derived",
  "body": "Erdős and Turán's 1941 paper is the classical source for \\(B_2\\) sequences defined by unique two-term sums. Bose and Chowla's 1962 paper develops finite cyclic \\(B_2\\) constructions. These sources treat a single additive group law.\n\nFor the multiplicative half of the present problem, discrete logarithms identify \\(\\mathbb F_{101}^{*}\\) with \\(\\mathbf Z/100\\mathbf Z\\). Buratti and Stinson define a \\((v,k)\\)-modular Golomb ruler by distinct directed differences modulo \\(v\\). Their Table 1 records nonexistence for \\(k=10\\) throughout \\(92\\leq v\\leq106\\), which includes \\(v=100\\). This supplies a published upper bound of 9 before the additive condition is imposed.\n\nThe focused audit found no primary source for the exact requirement that one subset of \\(\\mathbb F_{101}^{*}\\) be Sidon under both field addition and field multiplication. The claim of novelty therefore remains cautious. The exact answer here combines the published one-law obstruction with a directly checked simultaneous witness.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "primary literature on Sidon sets, finite cyclic B_2 constructions, and modular Golomb rulers checked through 2026-07-25",
    "bounds": {
      "field_order": {
        "min": 101,
        "max": 101
      },
      "literature_year": {
        "min": 1941,
        "max": 2021
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.26493/1855-3974.2374.9ff",
      "locator": "P. Erdős and P. Turán, On a Problem of Sidon in Additive Number Theory, and on Some Related Problems, Journal of the London Mathematical Society s1-16 (1941), 212-215, doi:10.1112/jlms/s1-16.4.212; R. C. Bose and S. Chowla, Theorems in the additive theory of numbers, Commentarii Mathematici Helvetici 37 (1962), 141-147, doi:10.1007/BF02566968; Marco Buratti and Douglas R. Stinson, Ars Mathematica Contemporanea 20 (2021), definition and Table 1"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.26493/1855-3974.2374.9ff",
    "locator": "P. Erdős and P. Turán, On a Problem of Sidon in Additive Number Theory, and on Some Related Problems, Journal of the London Mathematical Society s1-16 (1941), 212-215, doi:10.1112/jlms/s1-16.4.212; R. C. Bose and S. Chowla, Theorems in the additive theory of numbers, Commentarii Mathematici Helvetici 37 (1962), 141-147, doi:10.1007/BF02566968; Marco Buratti and Douglas R. Stinson, Ars Mathematica Contemporanea 20 (2021), definition and Table 1"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R736",
      "title": "The exact maximum is 9",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "simultaneous-sum-product-sidon-f101",
      "title": "simultaneous sum product sidon f101",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}