{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T02:44:35Z",
    "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": "2026-10-06T07:11:28+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:51:57+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q2313686",
    "registryId": "vr.tr.automorphic-number",
    "name": "automorphic number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q2313686/",
  "model": {
    "registry_id": "vr.tr.automorphic-number",
    "name": "automorphic number",
    "purpose": "Let an agent explain automorphic numbers, relay their definition, examples and construction from mathematical references, describe the connection to p-adic numbers, and distinguish them from related recreational number classes.",
    "definition": "A natural number whose square ends in the same digits as the number itself in a given base, such as 5, 6, 25, 76, 376 and 625 in base ten, arising from the solutions of n^2 = n modulo powers of the base and related to the 10-adic idempotents; automorphic numbers form two infinite sequences in base ten extending the digits ...90625 and ...09376, and generalise to trimorphic numbers whose cubes end in the same digits.",
    "what_it_is_for": "Not applicable; a mathematical concept.",
    "affordances": [
      "explain the definition",
      "relay examples and construction",
      "describe the p-adic connection",
      "distinguish related classes"
    ],
    "distinguishing_features": [
      "Self-reproducing square",
      "Base dependence",
      "Two infinite sequences",
      "p-adic idempotents"
    ],
    "appearance": "Not a visible object; integers with a self-reproducing square.",
    "visual_identification": [
      "Numbers whose square ends in the number",
      "Base ten sequences ending 90625 and 09376",
      "Trimorphic numbers use cubes; narcissistic numbers use digit powers"
    ],
    "physical_properties": [
      {
        "quantity": "first base ten examples",
        "typical_range": "0, 1, 5, 6, 25, 76, 376, 625",
        "unit": "sequence",
        "note": "OEIS A003226"
      },
      {
        "quantity": "base ten sequences",
        "typical_range": "2",
        "unit": "count",
        "note": "for n > 1 digits"
      }
    ],
    "families_and_kinds": [
      "automorphic numbers in base ten",
      "automorphic numbers in other bases",
      "trimorphic numbers",
      "circular and other digit-based number classes"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "non-negative integer",
        "why": "in registry terms"
      },
      {
        "relation": "is related to",
        "target": "p-adic number",
        "why": "through 10-adic idempotents"
      },
      {
        "relation": "is generalised by",
        "target": "trimorphic number",
        "why": "using cubes"
      },
      {
        "relation": "is listed in",
        "target": "OEIS",
        "why": "as A003226"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A003226",
        "note": "automorphic numbers"
      }
    ],
    "standards_and_regulation": [
      "No regulation; standard mathematical definitions"
    ],
    "failure_modes_and_hazards": [
      "Assuming base independence",
      "Confusing with other digit-based classes",
      "Miscounting sequences"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Explain what automorphic numbers are and list the known small ones in a stated base.",
        "Check or generate automorphic numbers in a given base and show the modular condition used."
      ],
      "must_not": [
        "State that a number is automorphic without saying in which base.",
        "Present a computational search result as a proof of a general pattern.",
        "Claim significance for these numbers in numerology, prediction or finance."
      ],
      "requires_human": [
        "Publishing a claimed new result about these numbers under a person's name."
      ]
    },
    "ethics": {
      "considerations": [
        "Number curiosities are often dressed up as mystical or predictive, which misleads people.",
        "Credit for computational results should go to those who produced them."
      ],
      "affected_parties": [
        "Students and learners",
        "Recreational mathematicians"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; integer sequence databases record the known values.",
      "master_systems": [
        "Published integer sequence databases"
      ]
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q2313686: automorphic number",
      "url": "https://www.wikidata.org/wiki/Q2313686",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Automorphic number",
      "url": "https://en.wikipedia.org/wiki/Automorphic_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 an automorphic number is.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "definition",
            "name": "Definition",
            "description": "Definition.",
            "findings": [
              {
                "id": "definition-finding",
                "name": "Definition",
                "description": "Definition.",
                "questions": [
                  {
                    "text": "What is an automorphic number, and what are the first examples in base ten?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about automorphic numbers or a related digit-based class?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "construction",
            "name": "Construction",
            "description": "Construction.",
            "findings": [
              {
                "id": "construction-finding",
                "name": "Construction",
                "description": "Construction.",
                "questions": [
                  {
                    "text": "How are longer automorphic numbers built digit by digit, and why are there exactly two sequences?",
                    "kind": "action"
                  },
                  {
                    "text": "Which entry fits the specific construction?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "theory",
        "name": "Theory",
        "description": "Theory.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "padic",
            "name": "p-adic",
            "description": "p-adic view.",
            "findings": [
              {
                "id": "padic-finding",
                "name": "p-adic",
                "description": "p-adic.",
                "questions": [
                  {
                    "text": "How do automorphic numbers correspond to idempotents in the 10-adic integers?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "bases",
            "name": "Bases",
            "description": "Other bases.",
            "findings": [
              {
                "id": "bases-finding",
                "name": "Bases",
                "description": "Bases.",
                "questions": [
                  {
                    "text": "How do automorphic numbers behave in bases that are prime powers or products of primes?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits number base?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "explore",
        "name": "Explore",
        "description": "Related classes.",
        "rationale": "Mathematics.",
        "layers": [
          {
            "id": "trimorphic",
            "name": "Trimorphic",
            "description": "Trimorphic numbers.",
            "findings": [
              {
                "id": "trimorphic-finding",
                "name": "Trimorphic",
                "description": "Trimorphic.",
                "questions": [
                  {
                    "text": "What are trimorphic numbers, and how do they relate?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which sources are cited?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "recreational",
            "name": "Recreational",
            "description": "Recreational mathematics.",
            "findings": [
              {
                "id": "recreational-finding",
                "name": "Recreational",
                "description": "Recreational.",
                "questions": [
                  {
                    "text": "How do automorphic numbers fit into recreational number theory?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits recreational mathematics?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and computation.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "When were automorphic numbers studied, and by whom?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the history of number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "computation",
            "name": "Computation",
            "description": "Computation.",
            "findings": [
              {
                "id": "computation-finding",
                "name": "Computation",
                "description": "Computation.",
                "questions": [
                  {
                    "text": "How can automorphic numbers be computed programmatically?",
                    "kind": "action"
                  },
                  {
                    "text": "Which entry fits computational number theory?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should trimorphic number be a separate entry?",
    "How should OEIS and references be linked?",
    "How should the p-adic interpretation be recorded?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}