{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-12T23:27:52Z",
    "providers": [
      "Claude"
    ],
    "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify",
    "pass": 2,
    "wave": 3,
    "engine": "claude",
    "enrichment": {
      "pass": 3,
      "engine": "claude",
      "producedAt": "2026-10-06T10:00:00Z",
      "ingestedAt": "2026-10-06T07:03:09+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:51:58+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q2915234",
    "registryId": "vr.tr.motzkin-number",
    "name": "Motzkin number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q2915234/",
  "model": {
    "registry_id": "vr.tr.motzkin-number",
    "name": "Motzkin number",
    "purpose": "Let an agent define Motzkin numbers, relay their combinatorial interpretations, recurrence and generating function, relate them to Catalan numbers, and help compute or verify terms.",
    "definition": "A number in the sequence 1, 1, 2, 4, 9, 21, 51, 127, 323, 835 and so on, counting the ways to draw non-crossing chords between some of n points on a circle, and equivalently counting Motzkin paths and other combinatorial objects; Motzkin numbers are named after Theodore Motzkin, satisfy a recurrence related to the Catalan numbers, and appear in enumerative combinatorics and computer science.",
    "what_it_is_for": "Counting combinatorial structures.",
    "affordances": [
      "define the sequence and interpretations",
      "relay recurrence and generating function",
      "relate to Catalan numbers",
      "compute or verify terms"
    ],
    "distinguishing_features": [
      "Non-crossing chord interpretation",
      "Motzkin path interpretation",
      "Three-term recurrence",
      "Algebraic generating function"
    ],
    "appearance": "Not a visible object; an integer sequence and lattice path diagrams.",
    "visual_identification": [
      "Sequence 1, 1, 2, 4, 9, 21, 51",
      "Counts non-crossing chords and Motzkin paths",
      "Catalan numbers count full non-crossing matchings; Schroder numbers are another related sequence"
    ],
    "physical_properties": [
      {
        "quantity": "first terms",
        "typical_range": "1, 1, 2, 4, 9, 21, 51, 127, 323, 835",
        "unit": "sequence",
        "note": ""
      }
    ],
    "families_and_kinds": [
      "Motzkin numbers",
      "Motzkin paths and their variants",
      "Motzkin triangle and refinements",
      "related sequences such as Catalan, Schroder and Riordan numbers"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "in registry terms"
      },
      {
        "relation": "is a kind of",
        "target": "integer sequence",
        "why": "in combinatorics"
      },
      {
        "relation": "is related to",
        "target": "Catalan number",
        "why": "by summation formula"
      },
      {
        "relation": "is named after",
        "target": "Theodore Motzkin",
        "why": "the mathematician"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A001006",
        "note": "Motzkin numbers"
      }
    ],
    "standards_and_regulation": [
      "No regulation; the OEIS is the reference database"
    ],
    "failure_modes_and_hazards": [
      "Confusing with Catalan numbers",
      "Indexing differences between sources",
      "Computation errors for large terms"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Compute Motzkin numbers and explain what they count.",
        "Cite the sequence from recognised integer sequence references."
      ],
      "must_not": [
        "Fabricate values, identities or proofs about the sequence.",
        "Present conjectures as theorems.",
        "Confuse Motzkin numbers with Catalan numbers or other sequences."
      ],
      "requires_human": [
        "Submitting proofs or results under a person's name."
      ]
    },
    "ethics": {
      "considerations": [
        "Mathematical credit belongs to those who did the work.",
        "Errors in published sequences spread to every later use."
      ],
      "affected_parties": [
        "Students and researchers",
        "Readers",
        "Authors of results"
      ]
    },
    "owners": {
      "steward": "Nobody owns a number sequence; it is held in common.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q2915234: Motzkin number",
      "url": "https://www.wikidata.org/wiki/Q2915234",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Motzkin number",
      "url": "https://en.wikipedia.org/wiki/Motzkin_number",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "define",
        "name": "Define",
        "description": "What Motzkin numbers are.",
        "rationale": "Definition.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition and interpretations.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What are Motzkin numbers, and which combinatorial objects do they count?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about Motzkin, Catalan or another related sequence?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "relations",
            "name": "Relations",
            "description": "Relation to Catalan numbers.",
            "findings": [
              {
                "id": "relations-finding",
                "name": "Relations",
                "description": "Relations.",
                "questions": [
                  {
                    "text": "How are Motzkin numbers expressed in terms of Catalan numbers?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which entry fits Catalan number?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "compute",
        "name": "Compute",
        "description": "Computation.",
        "rationale": "Computation.",
        "layers": [
          {
            "id": "recurrence",
            "name": "Recurrence",
            "description": "Recurrence and formulas.",
            "findings": [
              {
                "id": "recurrence-finding",
                "name": "Recurrence",
                "description": "Recurrence.",
                "questions": [
                  {
                    "text": "What recurrence, closed forms and generating function give Motzkin numbers?",
                    "kind": "action"
                  },
                  {
                    "text": "What is the term for the index in question?",
                    "kind": "measurement"
                  }
                ]
              }
            ]
          },
          {
            "id": "asymptotics",
            "name": "Asymptotics",
            "description": "Growth.",
            "findings": [
              {
                "id": "asymptotics-finding",
                "name": "Asymptotics",
                "description": "Asymptotics.",
                "questions": [
                  {
                    "text": "How fast do Motzkin numbers grow, and what is their asymptotic form?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "theory",
        "name": "Theory",
        "description": "Properties and generalisations.",
        "rationale": "Theory.",
        "layers": [
          {
            "id": "properties",
            "name": "Properties",
            "description": "Properties.",
            "findings": [
              {
                "id": "properties-finding",
                "name": "Properties",
                "description": "Properties.",
                "questions": [
                  {
                    "text": "What number-theoretic and combinatorial properties do Motzkin numbers have?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits enumerative combinatorics?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "generalisations",
            "name": "Generalisations",
            "description": "Generalisations.",
            "findings": [
              {
                "id": "generalisations-finding",
                "name": "Generalisations",
                "description": "Generalisations.",
                "questions": [
                  {
                    "text": "What generalisations and refinements exist, such as Motzkin triangles and coloured paths?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and applications.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How did Motzkin introduce the numbers in 1948, and how were interpretations discovered?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of combinatorics?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "applications",
            "name": "Applications",
            "description": "Applications.",
            "findings": [
              {
                "id": "applications-finding",
                "name": "Applications",
                "description": "Applications.",
                "questions": [
                  {
                    "text": "Where do Motzkin numbers appear in computer science and physics?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits lattice path?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should Motzkin path be a separate entry?",
    "How should the OEIS be linked?",
    "How should combinatorial interpretations be catalogued?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}