{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-11T07:21:51Z",
    "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": 1,
    "engine": "claude"
  },
  "metaModel": {
    "id": "THING-Q1130510",
    "registryId": "vr.tr.figurate-number",
    "name": "figurate number",
    "version": "0.2.0-wave.1",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q1130510/",
  "model": {
    "registry_id": "vr.tr.figurate-number",
    "name": "figurate number",
    "purpose": "Let an agent handle figurate numbers by shape, dimension, formula and sequence identifiers.",
    "definition": "A number that can be represented by a regular geometric arrangement of equally spaced points, such as triangular, square, pentagonal, centred and polyhedral numbers.",
    "what_it_is_for": "Number theory, combinatorics and recreational mathematics.",
    "affordances": [
      "compute the nth term",
      "test whether a number is figurate",
      "find sequences in OEIS",
      "visualise with dot patterns"
    ],
    "distinguishing_features": [
      "Numbers from geometric arrangements",
      "Polygonal, centred and polyhedral families",
      "Closed formulas",
      "Catalogued in OEIS"
    ],
    "appearance": "Dots arranged in triangles, squares, polygons or polyhedra.",
    "visual_identification": [
      "Dot patterns",
      "Names such as triangular or tetrahedral number",
      "Pronic numbers are products of consecutive integers"
    ],
    "physical_properties": [],
    "families_and_kinds": [
      "polygonal numbers",
      "centred polygonal numbers",
      "polyhedral numbers such as tetrahedral",
      "polytopic numbers such as pentatope",
      "pronic numbers"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "category"
      },
      {
        "relation": "relates to",
        "target": "polygon and polyhedron",
        "why": "shapes"
      },
      {
        "relation": "is catalogued in",
        "target": "OEIS",
        "why": "database"
      },
      {
        "relation": "is studied in",
        "target": "number theory",
        "why": "field"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS A-number",
        "value_or_pattern": "A000217 for triangular numbers",
        "note": "sequence ids"
      }
    ],
    "standards_and_regulation": [],
    "failure_modes_and_hazards": [
      "Off-by-one indexing conventions",
      "Confusing centred and plain forms"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": []
  },
  "sources": [],
  "structure": {
    "bundles": [
      {
        "id": "family",
        "name": "Family",
        "description": "Which figurate number.",
        "rationale": "Shape defines the family.",
        "layers": [
          {
            "id": "shape",
            "name": "Shape",
            "description": "Polygon or polyhedron.",
            "findings": [
              {
                "id": "shape-finding",
                "name": "Shape",
                "description": "Family.",
                "questions": [
                  {
                    "text": "Which shape and dimension define the family?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is it centred?",
                    "kind": "definition"
                  }
                ]
              }
            ]
          },
          {
            "id": "sequence",
            "name": "Sequence",
            "description": "OEIS.",
            "findings": [
              {
                "id": "sequence-finding",
                "name": "Sequence",
                "description": "Sequence.",
                "questions": [
                  {
                    "text": "What are the first terms?",
                    "kind": "measurement"
                  },
                  {
                    "text": "What is its OEIS number?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "formula",
        "name": "Formula",
        "description": "nth term.",
        "rationale": "Formulas compute terms.",
        "layers": [
          {
            "id": "term",
            "name": "Term",
            "description": "Closed form.",
            "findings": [
              {
                "id": "term-finding",
                "name": "Term",
                "description": "Formula.",
                "questions": [
                  {
                    "text": "What is the formula for the nth term?",
                    "kind": "definition"
                  },
                  {
                    "text": "Which index starts the sequence?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "test",
            "name": "Test",
            "description": "Membership.",
            "findings": [
              {
                "id": "test-finding",
                "name": "Test",
                "description": "Membership test.",
                "questions": [
                  {
                    "text": "Is this number in the family?",
                    "kind": "boundary"
                  },
                  {
                    "text": "How is that tested?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "relations",
        "name": "Relations",
        "description": "Identities.",
        "rationale": "Families are related.",
        "layers": [
          {
            "id": "identities",
            "name": "Identities",
            "description": "Sums.",
            "findings": [
              {
                "id": "identities-finding",
                "name": "Identities",
                "description": "Identities.",
                "questions": [
                  {
                    "text": "Which identities link families?",
                    "kind": "definition"
                  },
                  {
                    "text": "For example sums of triangular numbers?",
                    "kind": "definition"
                  }
                ]
              }
            ]
          },
          {
            "id": "theorems",
            "name": "Theorems",
            "description": "Results.",
            "findings": [
              {
                "id": "theorems-finding",
                "name": "Theorems",
                "description": "Theorems.",
                "questions": [
                  {
                    "text": "Which theorems apply, such as Fermat polygonal number theorem?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Who proved them?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "visual",
        "name": "Visual",
        "description": "Dot patterns.",
        "rationale": "Pictures explain.",
        "layers": [
          {
            "id": "drawing",
            "name": "Drawing",
            "description": "Patterns.",
            "findings": [
              {
                "id": "drawing-finding",
                "name": "Drawing",
                "description": "Dot patterns.",
                "questions": [
                  {
                    "text": "How can the pattern be drawn?",
                    "kind": "action"
                  },
                  {
                    "text": "How does it grow?",
                    "kind": "definition"
                  }
                ]
              }
            ]
          },
          {
            "id": "teaching",
            "name": "Teaching",
            "description": "Learning.",
            "findings": [
              {
                "id": "teaching-finding",
                "name": "Teaching",
                "description": "Teaching.",
                "questions": [
                  {
                    "text": "How can figurate numbers be taught?",
                    "kind": "action"
                  },
                  {
                    "text": "Which activities help?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should each family be a separate entry?",
    "How should OEIS ids be linked?",
    "How should indexing conventions be recorded?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}