{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R813",
  "content_hash": null,
  "slug": "twnfa-attempt-test-longer-property-chains",
  "type": "attempt",
  "title": "Test the proposed maximum length of the smooth-property chain",
  "summary": "Encode the Section 4 compatibility conditions as an exact finite search for \\(h=2,3,4\\), looking for a smooth-property chain longer than \\(\\binom{h+1}{2}\\) or a certificate that none exists in a stated matrix subclass.",
  "relevance": "For Polynomial determinization of two-way finite automata, record twnfa-attempt-test-longer-property-chains (“Test the proposed maximum length of the smooth-property chain”) documents a concrete method, search boundary, or failed route. The record states: Encode the Section 4 compatibility conditions as an exact finite search for \\(h=2,3,4\\), looking for a smooth-property chain longer than \\(\\binom{h+1}{2}\\) or a certificate that none exists in a stated matrix subclass.",
  "relevance_source": "recorded",
  "body": "Adeogun and Kapoutsis obtain their bound from a chain of \\(1+\\binom{h+1}{2}\\) smooth connectivity properties. Their conclusion conjectures that \\(\\binom{h+1}{2}\\) transitions are maximal for any chain satisfying the main lemma's successive-pair conditions.\n\nA useful next computation would formalize those conditions for connectivity properties \\(P(C)\\), enumerate or solve them with a certificate-producing SAT model for \\(h=2,3,4\\), and compare the exact maximum with the proposed value. The first run should restrict to idempotent matrices and should emit every adjacency witness used in the chain. A longer chain would sharpen this lower-bound method. Agreement at small \\(h\\) would delimit the method and provide test instances for a proof of maximality.",
  "status": "next_experiment",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "bounded",
    "statement": "certificate-producing search for connectivity-property chains at h=2, h=3, and h=4",
    "bounds": {
      "matrix_dimension_h": {
        "min": 2,
        "max": 4
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/2602.24279",
      "locator": "Proposed follow-up to Adeogun and Kapoutsis, Section 4 and the maximum-chain conjecture in Section 5"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2602.24279",
    "locator": "Proposed follow-up to Adeogun and Kapoutsis, Section 4 and the maximum-chain conjecture in Section 5"
  },
  "relations": [
    {
      "slug": "R818",
      "title": "One-way liveness forces at least h(h+1)/4 deterministic states",
      "object_type": "claim",
      "relation": "uses",
      "direction": "outgoing"
    },
    {
      "slug": "R815",
      "title": "The fixed-alphabet determinization question remains open",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "two-way-nfa-polynomial-determinization",
      "title": "two way nfa polynomial determinization",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}