{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-12T20:13:51Z",
    "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": 3,
    "engine": "claude",
    "enrichment": {
      "pass": 3,
      "engine": "claude",
      "producedAt": "2026-10-06T10:00:00Z",
      "ingestedAt": "2026-10-06T07:00:12+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:51:54+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q1462591",
    "registryId": "vr.tr.taxicab-number",
    "name": "taxicab number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q1462591/",
  "model": {
    "registry_id": "vr.tr.taxicab-number",
    "name": "taxicab number",
    "purpose": "Let an agent define taxicab numbers precisely, list the known values and their representations, distinguish them from related notions such as cabtaxi numbers, and relate them to the Hardy-Ramanujan anecdote and to Diophantine problems.",
    "definition": "The smallest positive integer that can be expressed as a sum of two positive cubes in n distinct ways, written Ta(n); the first taxicab numbers are 2 for n equal to 1, 1729 for n equal to 2, and 87539319 for n equal to 3, and the name recalls the anecdote of Hardy and Ramanujan about the taxi numbered 1729.",
    "what_it_is_for": "Not for anything; a curiosity in number theory.",
    "affordances": [
      "state the definition and known values",
      "verify a representation as a sum of two cubes",
      "distinguish taxicab from cabtaxi and generalised numbers",
      "point to sequence references"
    ],
    "distinguishing_features": [
      "Defined by sums of two positive cubes",
      "Smallest with n representations",
      "Only a few values known exactly",
      "Named for the Hardy-Ramanujan anecdote"
    ],
    "appearance": "Not visible; a property of integers.",
    "visual_identification": [
      "Smallest number with n representations as a sum of two positive cubes",
      "Ta(2) equals 1729 equals 1 cubed plus 12 cubed equals 9 cubed plus 10 cubed",
      "Cabtaxi numbers allow negative cubes and are a different sequence"
    ],
    "physical_properties": [
      {
        "quantity": "Ta(2)",
        "typical_range": "1729",
        "unit": "integer",
        "note": ""
      },
      {
        "quantity": "Ta(3)",
        "typical_range": "87539319",
        "unit": "integer",
        "note": ""
      },
      {
        "quantity": "known values",
        "typical_range": "Ta(1) to Ta(6)",
        "unit": "count",
        "note": "later values have upper bounds only"
      }
    ],
    "families_and_kinds": [
      "taxicab numbers Ta(n)",
      "cabtaxi numbers allowing negative cubes",
      "generalised taxicab numbers for other powers and more terms",
      "upper bounds for unknown values"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "a property of integers"
      },
      {
        "relation": "is related to",
        "target": "1729",
        "why": "the famous instance"
      },
      {
        "relation": "is confused with",
        "target": "cabtaxi number",
        "why": "a related sequence allowing negative cubes"
      },
      {
        "relation": "is listed in",
        "target": "On-Line Encyclopedia of Integer Sequences",
        "why": "sequence A011541"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A011541",
        "note": "taxicab numbers"
      }
    ],
    "standards_and_regulation": [
      "No regulation; definitions follow the literature and the OEIS"
    ],
    "failure_modes_and_hazards": [
      "Quoting upper bounds as exact values",
      "Confusing taxicab and cabtaxi numbers",
      "Misstating the anecdote"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Explain taxicab numbers and the known values from published sources.",
        "Verify that a number has the stated representations as sums of cubes."
      ],
      "must_not": [
        "Present an upper bound as a proven taxicab number when it is not proven minimal.",
        "Misstate the story of 1729 or attribute it to the wrong people.",
        "Confuse taxicab numbers with cabtaxi numbers or with taxicab geometry."
      ],
      "requires_human": [
        "Correcting a published value."
      ]
    },
    "ethics": {
      "considerations": [
        "Accurate records of mathematical results protect credit and trust.",
        "Popular anecdotes are often repeated with errors."
      ],
      "affected_parties": [
        "Students",
        "Mathematicians"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; it belongs to mathematics.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q1462591: taxicab number",
      "url": "https://www.wikidata.org/wiki/Q1462591",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Taxicab number",
      "url": "https://en.wikipedia.org/wiki/Taxicab_number",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "define",
        "name": "Define",
        "description": "The definition.",
        "rationale": "Precision.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What exactly is Ta(n), and what counts as a distinct representation?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about taxicab numbers, cabtaxi numbers or a generalisation?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "values",
            "name": "Values",
            "description": "Known values.",
            "findings": [
              {
                "id": "values-finding",
                "name": "Values",
                "description": "Values.",
                "questions": [
                  {
                    "text": "What are the known taxicab numbers and their representations?",
                    "kind": "measurement"
                  },
                  {
                    "text": "Which values are exact and which are bounds?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "compute",
        "name": "Compute",
        "description": "Computation.",
        "rationale": "Computation.",
        "layers": [
          {
            "id": "verify",
            "name": "Verify",
            "description": "Verification.",
            "findings": [
              {
                "id": "verify-finding",
                "name": "Verify",
                "description": "Verify.",
                "questions": [
                  {
                    "text": "How is a claimed representation verified, and how are candidates searched?",
                    "kind": "action"
                  },
                  {
                    "text": "Which entry fits Diophantine equations?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "bounds",
            "name": "Bounds",
            "description": "Bounds.",
            "findings": [
              {
                "id": "bounds-finding",
                "name": "Bounds",
                "description": "Bounds.",
                "questions": [
                  {
                    "text": "How were bounds for higher taxicab numbers found, with attribution?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "theory",
        "name": "Theory",
        "description": "Mathematics behind it.",
        "rationale": "Theory.",
        "layers": [
          {
            "id": "cubes",
            "name": "Cubes",
            "description": "Sums of two cubes.",
            "findings": [
              {
                "id": "cubes-finding",
                "name": "Cubes",
                "description": "Cubes.",
                "questions": [
                  {
                    "text": "What is known about representing numbers as sums of two cubes, and why do multiple representations exist?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits sums of powers?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "generalise",
            "name": "Generalise",
            "description": "Generalisations.",
            "findings": [
              {
                "id": "generalise-finding",
                "name": "Generalise",
                "description": "Generalise.",
                "questions": [
                  {
                    "text": "What generalisations to other powers and more terms exist?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Should a generalisation be a separate entry?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and culture.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "anecdote",
            "name": "Anecdote",
            "description": "Hardy and Ramanujan.",
            "findings": [
              {
                "id": "anecdote-finding",
                "name": "Anecdote",
                "description": "Anecdote.",
                "questions": [
                  {
                    "text": "What is the Hardy-Ramanujan anecdote, and how is it documented?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits Ramanujan?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "teach",
            "name": "Teach",
            "description": "Teaching.",
            "findings": [
              {
                "id": "teach-finding",
                "name": "Teach",
                "description": "Teaching.",
                "questions": [
                  {
                    "text": "How are taxicab numbers used in teaching and popular mathematics?",
                    "kind": "action"
                  },
                  {
                    "text": "Which misconceptions arise?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Which values beyond Ta(6) can be recorded as bounds?",
    "How should OEIS entries be linked?",
    "Should 1729 be a separate entry?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}