{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T03:05:08Z",
    "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": "2026-10-06T07:06:07+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:52:00+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q4516348",
    "registryId": "vr.tr.cake-number",
    "name": "cake number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q4516348/",
  "model": {
    "registry_id": "vr.tr.cake-number",
    "name": "cake number",
    "purpose": "Let an agent explain cake numbers and their formula, relay the derivation and related sequences from mathematical references, describe puzzle applications, and distinguish cake numbers from the lazy caterer sequence and other cutting problems.",
    "definition": "The maximum number of regions into which a three-dimensional cube or cake can be divided by n planar cuts, given by the formula (n^3 + 5n + 6) / 6, producing the sequence 1, 2, 4, 8, 15, 26, 42, 64 and so on, the three-dimensional analogue of the lazy caterer sequence for cutting a circle with lines; cake numbers are a classic example in combinatorial geometry and appear in puzzles about cutting a cake into the most pieces.",
    "what_it_is_for": "Not applicable; a mathematical concept.",
    "affordances": [
      "explain the definition and formula",
      "relay the derivation",
      "describe puzzle uses",
      "distinguish related sequences"
    ],
    "distinguishing_features": [
      "Cubic growth",
      "Binomial sum form",
      "Three-dimensional cutting",
      "Puzzle origin"
    ],
    "appearance": "Not a visible object; a cube divided by planes.",
    "visual_identification": [
      "Maximum pieces from n planar cuts of a cube",
      "Sequence 1, 2, 4, 8, 15, 26, 42, 64",
      "The lazy caterer sequence is the two-dimensional case; powers of two count pieces only up to three cuts"
    ],
    "physical_properties": [
      {
        "quantity": "formula",
        "typical_range": "(n^3 + 5n + 6) / 6",
        "unit": "formula",
        "note": ""
      },
      {
        "quantity": "first terms",
        "typical_range": "1, 2, 4, 8, 15, 26, 42, 64, 93",
        "unit": "sequence",
        "note": "OEIS A000125"
      },
      {
        "quantity": "binomial form",
        "typical_range": "C(n,3) + C(n,2) + C(n,1) + C(n,0)",
        "unit": "formula",
        "note": ""
      }
    ],
    "families_and_kinds": [
      "cake numbers for planes in three dimensions",
      "lazy caterer numbers for lines in two dimensions",
      "higher-dimensional generalisations by hyperplanes",
      "related problems such as cutting with restricted cuts"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "in registry terms"
      },
      {
        "relation": "generalises",
        "target": "lazy caterer sequence",
        "why": "to three dimensions"
      },
      {
        "relation": "is listed in",
        "target": "OEIS",
        "why": "as A000125"
      },
      {
        "relation": "is studied in",
        "target": "combinatorial geometry",
        "why": "and recreational mathematics"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A000125",
        "note": "cake numbers"
      }
    ],
    "standards_and_regulation": [
      "No regulation; standard mathematical definitions"
    ],
    "failure_modes_and_hazards": [
      "Assuming pieces double with each cut",
      "Confusing with the lazy caterer sequence",
      "Requiring equal pieces, which the problem does not"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Compute cake numbers from the formula and check them against the known sequence.",
        "Use the cake number problem to teach counting regions and dimensions."
      ],
      "must_not": [
        "Confuse the cake number with the lazy caterer's number for the plane.",
        "Present wrong terms or a wrong formula as established.",
        "Treat the name as describing actual cake cutting or portion sizes."
      ],
      "requires_human": [
        "Publishing corrections to shared reference entries for the sequence."
      ]
    },
    "ethics": {
      "considerations": [
        "Teaching material should be accurate because students rely on it.",
        "Credit for the result belongs to those who established it."
      ],
      "affected_parties": [
        "Students and teachers",
        "Puzzle readers"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; it is held in common.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q4516348: cake number",
      "url": "https://www.wikidata.org/wiki/Q4516348",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Cake number",
      "url": "https://en.wikipedia.org/wiki/Cake_number",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "understand",
        "name": "Understand",
        "description": "What a cake number is.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is a cake number, and what are the first terms of the sequence?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about cake numbers, the lazy caterer sequence or another cutting problem?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "formula",
            "name": "Formula",
            "description": "Formula.",
            "findings": [
              {
                "id": "formula-finding",
                "name": "Formula",
                "description": "Formula.",
                "questions": [
                  {
                    "text": "How is the formula (n^3 + 5n + 6) / 6 derived, and why does it equal a sum of binomial coefficients?",
                    "kind": "action"
                  },
                  {
                    "text": "Which entry fits binomial coefficient?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "explore",
        "name": "Explore",
        "description": "Related problems.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "twod",
            "name": "Two dimensions",
            "description": "Lazy caterer.",
            "findings": [
              {
                "id": "twod-finding",
                "name": "Two dimensions",
                "description": "Two dimensions.",
                "questions": [
                  {
                    "text": "How does the two-dimensional lazy caterer sequence relate, and how do the formulas generalise to higher dimensions?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "constraints",
            "name": "Constraints",
            "description": "Constrained cuts.",
            "findings": [
              {
                "id": "constraints-finding",
                "name": "Constraints",
                "description": "Constraints.",
                "questions": [
                  {
                    "text": "What happens if cuts must pass through a point or produce equal pieces?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "apply",
        "name": "Apply",
        "description": "Puzzles and teaching.",
        "rationale": "Practice.",
        "layers": [
          {
            "id": "puzzles",
            "name": "Puzzles",
            "description": "Puzzles.",
            "findings": [
              {
                "id": "puzzles-finding",
                "name": "Puzzles",
                "description": "Puzzles.",
                "questions": [
                  {
                    "text": "How do cake-cutting puzzles use cake numbers, and what are common wrong answers?",
                    "kind": "action"
                  },
                  {
                    "text": "Which entry fits recreational mathematics?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "teaching",
            "name": "Teaching",
            "description": "Teaching.",
            "findings": [
              {
                "id": "teaching-finding",
                "name": "Teaching",
                "description": "Teaching.",
                "questions": [
                  {
                    "text": "How can cake numbers be used to teach recurrences and combinatorics?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits mathematics education?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "Where did the problem originate, and who studied hyperplane arrangements?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits hyperplane arrangement?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "computation",
            "name": "Computation",
            "description": "Computation.",
            "findings": [
              {
                "id": "computation-finding",
                "name": "Computation",
                "description": "Computation.",
                "questions": [
                  {
                    "text": "How can cake numbers be computed and verified programmatically?",
                    "kind": "action"
                  },
                  {
                    "text": "Which entry fits integer sequence?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should the lazy caterer sequence be a separate entry?",
    "How should OEIS and references be linked?",
    "How should higher-dimensional generalisations be recorded?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}