{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-12T20:28:17Z",
    "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": "2026-10-06T06:57:49+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-Q1529876",
    "registryId": "vr.tr.smooth-number",
    "name": "smooth number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q1529876/",
  "model": {
    "registry_id": "vr.tr.smooth-number",
    "name": "smooth number",
    "purpose": "Let an agent define smooth numbers precisely, test smoothness for a given bound, explain their distribution and their role in algorithms, and relate regular numbers to music, computing and history.",
    "definition": "A positive integer whose prime factors are all at most some bound B, called B-smooth; for example 5-smooth or regular numbers have only the factors 2, 3 and 5; smooth numbers are central to factoring and discrete logarithm algorithms, to fast Fourier transform sizes and to ancient Babylonian arithmetic.",
    "what_it_is_for": "Not for anything in itself; a property used in algorithms and computation.",
    "affordances": [
      "test whether a number is B-smooth",
      "explain the definition and examples",
      "explain the role in factoring algorithms",
      "relate regular numbers to their uses"
    ],
    "distinguishing_features": [
      "Bounded largest prime factor",
      "Density described by the Dickman function",
      "Key to sieve-based algorithms",
      "Regular numbers as the classic case"
    ],
    "appearance": "Not visible; a property of integers.",
    "visual_identification": [
      "All prime factors at most B",
      "Regular numbers are 5-smooth: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12",
      "Rough numbers have only large prime factors, the opposite property"
    ],
    "physical_properties": [
      {
        "quantity": "regular numbers below 60",
        "typical_range": "1 to 60 with factors 2, 3, 5 only",
        "unit": "integers",
        "note": "used in Babylonian reciprocal tables"
      }
    ],
    "families_and_kinds": [
      "B-smooth numbers for a bound B",
      "regular or 5-smooth numbers",
      "7-smooth humble numbers",
      "powersmooth numbers",
      "smooth numbers in algorithm analysis"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "a property of integers"
      },
      {
        "relation": "is used in",
        "target": "integer factorization",
        "why": "sieve algorithms"
      },
      {
        "relation": "is contrasted with",
        "target": "rough number",
        "why": "the opposite property"
      },
      {
        "relation": "is listed in",
        "target": "On-Line Encyclopedia of Integer Sequences",
        "why": "sequence A051037 for regular numbers"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A051037",
        "note": "regular numbers"
      }
    ],
    "standards_and_regulation": [
      "No regulation; definitions follow the literature"
    ],
    "failure_modes_and_hazards": [
      "Omitting the bound B when calling a number smooth",
      "Confusing smooth with rough or with squarefree",
      "Presenting heuristic distribution results as exact"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "may": [
        "Test whether a number is B-smooth and state the bound used.",
        "Explain the role of smooth numbers in factoring, discrete logarithms and transform sizes."
      ],
      "must_not": [
        "Call a number smooth without stating the bound.",
        "Confuse smooth numbers with rough or squarefree numbers.",
        "Present heuristic estimates of how many smooth numbers exist as proven results.",
        "Overstate what smooth-number methods imply for the security of deployed cryptography."
      ],
      "requires_human": [
        "Choosing cryptographic key sizes on the basis of smoothness estimates."
      ]
    },
    "ethics": {
      "considerations": [
        "Results about smooth numbers bear on how secure some cryptographic systems are."
      ],
      "affected_parties": [
        "Learners of number theory",
        "People relying on cryptography"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; it belongs to the shared body of mathematics.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q1529876: smooth number",
      "url": "https://www.wikidata.org/wiki/Q1529876",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Smooth number",
      "url": "https://en.wikipedia.org/wiki/Smooth_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 does B-smooth mean, and what are regular numbers?",
                    "kind": "definition"
                  },
                  {
                    "text": "Is the question about smooth, rough or squarefree numbers?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          },
          {
            "id": "test",
            "name": "Test",
            "description": "Testing.",
            "findings": [
              {
                "id": "test-finding",
                "name": "Test",
                "description": "Test.",
                "questions": [
                  {
                    "text": "How is smoothness tested for a given number and bound?",
                    "kind": "action"
                  },
                  {
                    "text": "What is the result for the number in question?",
                    "kind": "measurement"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "distribution",
        "name": "Distribution",
        "description": "How common they are.",
        "rationale": "Theory.",
        "layers": [
          {
            "id": "density",
            "name": "Density",
            "description": "Density.",
            "findings": [
              {
                "id": "density-finding",
                "name": "Density",
                "description": "Density.",
                "questions": [
                  {
                    "text": "How many smooth numbers are there up to x, and what does the Dickman function describe?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which references are standard?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          },
          {
            "id": "results",
            "name": "Results",
            "description": "Known results.",
            "findings": [
              {
                "id": "results-finding",
                "name": "Results",
                "description": "Results.",
                "questions": [
                  {
                    "text": "What is proven versus heuristic about smooth number distribution?",
                    "kind": "provenance"
                  },
                  {
                    "text": "What is proven versus conjectured?",
                    "kind": "boundary"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "algorithms",
        "name": "Algorithms",
        "description": "Uses in computation.",
        "rationale": "Applications.",
        "layers": [
          {
            "id": "factoring",
            "name": "Factoring",
            "description": "Factoring and logarithms.",
            "findings": [
              {
                "id": "factoring-finding",
                "name": "Factoring",
                "description": "Factoring.",
                "questions": [
                  {
                    "text": "How do the quadratic sieve, number field sieve and index calculus use smooth numbers?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits integer factorization?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "other",
            "name": "Other",
            "description": "Other uses.",
            "findings": [
              {
                "id": "other-finding",
                "name": "Other",
                "description": "Other.",
                "questions": [
                  {
                    "text": "How are regular numbers used in FFT sizes, music theory and Babylonian tables?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits the fast Fourier transform?",
                    "kind": "action"
                  }
                ]
              }
            ]
          }
        ]
      },
      {
        "id": "context",
        "name": "Context",
        "description": "History and teaching.",
        "rationale": "Context.",
        "layers": [
          {
            "id": "history",
            "name": "History",
            "description": "History.",
            "findings": [
              {
                "id": "history-finding",
                "name": "History",
                "description": "History.",
                "questions": [
                  {
                    "text": "How did regular numbers appear in Babylonian mathematics and in later theory?",
                    "kind": "provenance"
                  },
                  {
                    "text": "Which entry fits Babylonian mathematics?",
                    "kind": "action"
                  }
                ]
              }
            ]
          },
          {
            "id": "teach",
            "name": "Teach",
            "description": "Teaching.",
            "findings": [
              {
                "id": "teach-finding",
                "name": "Teach",
                "description": "Teaching.",
                "questions": [
                  {
                    "text": "How are smooth numbers taught in number theory and cryptography courses?",
                    "kind": "action"
                  },
                  {
                    "text": "Which misconceptions arise?",
                    "kind": "provenance"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should regular numbers be a separate entry?",
    "How should OEIS entries be linked?",
    "How should the Dickman function be represented?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}