{ "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-12T09:37:18Z", "providers": [ "Codex" ], "breadth": "recalled by Codex without web access - no source was read", "missingProviders": [], "pass": 2, "cost": { "grok": { "seconds": 14.0, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.4\n--------\nworkdir: R:\\02_PROJECTS\\02_Meta_Models_Platforms\\Ver.cy\\current\\thing-registry-backlog\nmodel: gpt-6-astra\nprovider: openai\napproval: never\nsandbox: read-only\nreasoning effort: none\nreasoning summaries: none\nsession id: 01a094f9-4545-72a1-8827-9e8dde22dd51\n--------\nuser\nDescribe what is already known about one registered thing. Answer as JSON only, no prose around it.\n\nThing: factorion\nSense to describe: (none recorded)\nDomain code: XCT.QTY\nAlso known as: (none)\n\n\nContext for this batch of 1001 things:\n# Batch 043: 1001 regis", "usd": 0.0, "recall": true }, "codex": { "seconds": 79.1, "error": "Reading additional input from stdin...\nOpenAI Codex v0.153.4\n--------\nworkdir: R:\\02_PROJECTS\\02_Meta_Models_Platforms\\Ver.cy\\current\\thing-registry-backlog\nmodel: gpt-6-astra\nprovider: openai\napproval: never\nsandbox: read-only\nreasoning effort: none\nreasoning summaries: none\nsession id: 01a094f9-4553-7550-8061-3fa5bb8900aa\n--------\nuser\nYou are drafting a Vercy meta-model for one registered thing. Answer as JSON only, no prose around it.\n\nThing: factorion\nRegistry id: vr.tr.factorion\nPlane / domain: XCT / XCT.QTY\nRegistry definition: (none recorded)\nNames folded into this entry: (none)\n\n\nCont" } }, "enrichment": { "pass": 3, "engine": "claude", "producedAt": "2026-10-06T11:00:00Z", "ingestedAt": "2026-10-06T08:15:58+00:00", "fields": [ "agent_conduct", "ethics", "failure_modes_and_hazards", "owners" ] }, "revisions": [ { "at": "2026-10-07T11:22:46+00:00", "engine": "codex", "fields": [ "agent_conduct", "definition", "distinguishing_features", "in_scope", "purpose", "scope_statement" ], "answering_review": "189006e0a525c564c8247d92a89d19ed955b945a056ef90e747b4960d0280481" } ], "gate": { "passed": true, "version": "1.0", "at": "2026-10-07T12:16:30+00:00", "evidence": [ "completeness 1.00", "sense: Wikidata description", "2 live sources", "review by codex: pass" ] } }, "metaModel": { "id": "THING-Q1977759", "registryId": "vr.tr.factorion", "name": "factorion", "version": "0.1.0-research.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q1977759/", "model": { "registry_id": "vr.tr.factorion", "name": "factorion", "purpose": "Enable an agent to recognise a factorion, verify its decimal digit-factorial equality, and judge whether a claimed enumeration is complete.", "definition": "A factorion is a positive integer equal to the sum of the factorials of its decimal digits.", "scope_statement": "This model covers positive integers satisfying n = sum(d!), where d ranges over every digit occurrence in the canonical decimal representation of n.", "in_scope": [ "The decimal digit-factorial equality defines membership.", "Canonical decimal digits and the treatment of zero digits determine the sum.", "Exact verification establishes whether an individual candidate is a factorion.", "Finite search bounds support completeness claims for enumerations.", "Factorions are distinguished from other digit-defined number classes." ], "out_of_scope": [ "Factorials as a general mathematical operation", "Factorial or factoradic numeral systems", "Narcissistic numbers defined using digit powers", "General dynamics of digit-factorial chains beyond their fixed points", "Physical quantities, measurement units, and conversion systems" ], "distinguishing_features": [ "The integer equals the sum of factorials of its own digits, rather than the factorial of the integer.", "Membership in this model is evaluated using decimal digits, while analogous properties in other bases are separate generalisations.", "Every digit occurrence contributes separately, including zero because 0! = 1.", "A factorion is a fixed point of the digit-factorial map; merely reaching a fixed point after several iterations does not establish membership.", "Digit factorials determine the sum, distinguishing the test from digit-power tests for narcissistic numbers." ], "characteristics": [ { "name": "Candidate integer", "kind": "measurement", "unit_or_values": "Exact positive integer; no physical unit", "why_it_matters": "Identifies the value whose membership is being tested." }, { "name": "Numeral base", "kind": "category", "unit_or_values": "Integer b >= 2; decimal convention b = 10", "why_it_matters": "Determines the digits used in the defining equality." }, { "name": "Canonical digit sequence", "kind": "relation", "unit_or_values": "Digits in 0 through b - 1, with no leading zeros", "why_it_matters": "Connects the integer to the representation used for verification." }, { "name": "Digit multiplicities", "kind": "measurement", "unit_or_values": "Count c_d >= 0 for each allowed digit d", "why_it_matters": "Supports computation of sum(c_d * d!) and searches over digit multisets." }, { "name": "Digit-factorial sum", "kind": "measurement", "unit_or_values": "Exact integer S_b(n) = sum(d!)", "why_it_matters": "Provides the value compared with the candidate." }, { "name": "Equality residual", "kind": "measurement", "unit_or_values": "Exact integer S_b(n) - n", "why_it_matters": "Zero establishes membership when the input representation is valid." }, { "name": "Verification state", "kind": "state", "unit_or_values": "Untested | verified member | verified nonmember | invalid input", "why_it_matters": "Separates mathematical membership from the agent's evidence about it." }, { "name": "Enumeration coverage", "kind": "state", "unit_or_values": "Examples only | bounded search completed | exhaustive enumeration justified", "why_it_matters": "Prevents a list of successful candidates from being mistaken for all factorions." } ], "affordances": [ "Verify a candidate using exact digit extraction and factorial arithmetic.", "Produce an inspectable certificate listing digits, factorial contributions, and their sum.", "Re-evaluate membership after changing the numeral base.", "Enumerate candidates within a justified finite bound.", "Reject claims based on padded numerals, digit powers, or nontrivial iteration cycles.", "Compare a claimed complete list with its proof and search coverage." ], "agent_conduct": { "may": [ "Find, verify and explain decimal factorions.", "Explain why only finitely many decimal factorions exist." ], "must_not": [ "Report a number as a factorion in this model based only on a nondecimal representation.", "Claim a complete list of decimal factorions without a proof or exhaustive check.", "Confuse factorions with other digit-based curiosities such as narcissistic numbers." ], "requires_human": [ "Publishing new results under someone else's name requires human involvement." ] }, "ethics": { "considerations": [ "Recreational mathematics invites learners in, so wrong claims mislead them." ], "affected_parties": [ "Students", "Puzzle enthusiasts" ] }, "owners": { "steward": "Nobody owns them; they are held in common as mathematics.", "master_systems": [] }, "failure_modes_and_hazards": [ "Errors from forgetting that 0! equals 1.", "Mixing results across numeral bases." ], "what_it_is_for": [ "Factorions are studied in recreational mathematics and elementary number theory as examples of integers defined by their digits.", "They are used in programming exercises involving positional notation, factorials and exhaustive searches over bounded sets of integers." ], "facets_not_applicable": { "interfaces": "No specific standard, law or official classification governs factorions as mathematical objects.", "recognition": "A factorion is an abstract integer property established by calculation rather than visual appearance." } }, "sources": [ { "id": "wikidata", "title": "Wikidata item Q1977759: factorion", "url": "https://www.wikidata.org/wiki/Q1977759", "url_status": "live", "what_it_supports": "identity and sense of the item", "checked_by": "wikidata-api" }, { "id": "wikipedia-en", "title": "Wikipedia: Factorion", "url": "https://en.wikipedia.org/wiki/Factorion", "url_status": "live", "what_it_supports": "general description of the item", "checked_by": "wikidata-api sitelink" } ], "structure": { "bundles": [ { "id": "factorion-membership", "name": "Factorion membership", "description": "The defining equality and the numerical domain in which it applies.", "rationale": "An agent needs a precise membership predicate before accepting examples or classifications.", "layers": [ { "id": "defining-equality", "name": "Defining equality", "description": "Specify the operation applied to the candidate's digits.", "findings": [ { "id": "digit-factorial-fixed-point", "name": "Digit-factorial fixed point", "description": "A factorion equals the sum of the factorials of its digits. [MathWorld: Factorion](https://mathworld.wolfram.com/Factorion.html)", "questions": [ { "text": "Does the candidate satisfy n = S_b(n) using its own base-b digits?", "kind": "definition", "id": "digit-factorial-fixed-point-q01" }, { "text": "Does the claim apply factorial to each digit separately before summing?", "kind": "boundary", "id": "digit-factorial-fixed-point-q02" } ] } ] }, { "id": "admissible-values", "name": "Admissible values", "description": "Separate valid integer candidates from inputs outside the definition.", "findings": [ { "id": "integer-domain-and-zero", "name": "Integer domain and zero", "description": "Use positive integer candidates; if zero is tested as an extension of the input domain, its canonical digit contributes 0! = 1, so it fails the equality.", "questions": [ { "text": "Is the supplied candidate an exact positive integer?", "kind": "boundary", "id": "integer-domain-and-zero-q01" }, { "text": "Does the verifier correctly reject zero, negative values, and nonintegers as members?", "kind": "action", "id": "integer-domain-and-zero-q02" } ] } ] } ] }, { "id": "base-and-digit-semantics", "name": "Base and digit semantics", "description": "The representation choices that determine the factorial sum.", "rationale": "Factorion membership depends on digits, so an unstated or malformed representation can change the result.", "layers": [ { "id": "positional-base", "name": "Positional base", "description": "Bind each membership claim to a standard positional integer base.", "findings": [ { "id": "base-qualified-membership", "name": "Base-qualified membership", "description": "The definition extends to specified positional integer bases; membership should therefore be attached to an integer and base together. [PlanetMath: factorion](https://planetmath.org/Factorion)", "questions": [ { "text": "Which integer base b >= 2 governs this claim, and was it stated or assumed?", "kind": "definition", "id": "base-qualified-membership-q01" }, { "text": "When changing base, has the agent preserved the integer value and recomputed its digits?", "kind": "action", "id": "base-qualified-membership-q02" } ] } ] }, { "id": "canonical-digit-handling", "name": "Canonical digit handling", "description": "Preserve legitimate zero and repeated digits while excluding representational padding.", "findings": [ { "id": "zero-and-multiplicity", "name": "Zero and multiplicity", "description": "Record each digit occurrence, retain internal and trailing zeros, and exclude leading padding; adding a leading zero would spuriously add 1 to the factorial sum.", "questions": [ { "text": "Does the recorded digit sequence reconstruct n without leading zeros?", "kind": "measurement", "id": "zero-and-multiplicity-q01" }, { "text": "Are repeated digits counted separately and all genuine zero digits assigned factorial value 1?", "kind": "measurement", "id": "zero-and-multiplicity-q02" } ] } ] } ] }, { "id": "exact-membership-verification", "name": "Exact membership verification", "description": "Evidence sufficient to accept or reject an individual candidate.", "rationale": "A claimed factorion must be checkable without trusting a label or an opaque computation.", "layers": [ { "id": "arithmetic-certificate", "name": "Arithmetic certificate", "description": "Expose the complete digit-factorial calculation.", "findings": [ { "id": "recomputable-equality", "name": "Recomputable equality", "description": "Record the base, canonical digits, individual factorial contributions, exact sum, and residual against n.", "questions": [ { "text": "What are the exact factorial contributions and their total?", "kind": "measurement", "id": "recomputable-equality-q01" }, { "text": "Can an independent calculation reproduce the zero or nonzero residual without rounding or overflow?", "kind": "action", "id": "recomputable-equality-q02" } ] } ] }, { "id": "decimal-reference-cases", "name": "Decimal reference cases", "description": "Use established decimal members to check the implementation and interpretation.", "findings": [ { "id": "known-decimal-members", "name": "Known decimal members", "description": "The decimal factorions are 1, 2, 145, and 40585; 145 = 1! + 4! + 5!, while 40585 exercises both a zero digit and a repeated digit. [MathWorld: Factorion](https://mathworld.wolfram.com/Factorion.html)", "questions": [ { "text": "Does the verifier accept all four reference members, including the single-digit cases?", "kind": "action", "id": "known-decimal-members-q01" }, { "text": "Which exact calculation supports each accepted reference case and each chosen nonmember test?", "kind": "measurement", "id": "known-decimal-members-q02" } ] } ] } ] }, { "id": "finite-enumeration", "name": "Finite enumeration", "description": "Search limits, candidate generation, and evidence that no members were omitted.", "rationale": "Verifying examples and proving a complete catalogue require different evidence.", "layers": [ { "id": "digit-length-bound", "name": "Digit-length bound", "description": "Bound possible solutions by comparing positional magnitude with factorial contributions.", "findings": [ { "id": "necessary-size-inequality", "name": "Necessary size inequality", "description": "A derived necessary condition for a k-digit base-b factorion is b^(k-1) <= k * (b-1)!: the left side bounds its value below and the right side bounds its digit-factorial sum above.", "questions": [ { "text": "Which digit lengths satisfy this necessary inequality for the selected base?", "kind": "measurement", "id": "necessary-size-inequality-q01" }, { "text": "What argument establishes that every length beyond the chosen cutoff also fails the inequality?", "kind": "definition", "id": "necessary-size-inequality-q02" } ] } ] }, { "id": "search-completeness", "name": "Search completeness", "description": "Record how the finite candidate space was exhausted.", "findings": [ { "id": "coverage-and-reconstruction", "name": "Coverage and reconstruction", "description": "A complete search must cover every admissible candidate or digit multiset; multiset searches must confirm that each computed sum has exactly the digits hypothesised.", "questions": [ { "text": "What search range, generation method, and pruning arguments establish complete coverage?", "kind": "provenance", "id": "coverage-and-reconstruction-q01" }, { "text": "For a digit-multiset search, does each retained sum reproduce the same base-b digit multiplicities?", "kind": "action", "id": "coverage-and-reconstruction-q02" } ] } ] } ] }, { "id": "neighbouring-number-classes", "name": "Neighbouring number classes", "description": "Boundaries against similar factorial and digit-based constructions.", "rationale": "Similar terminology and related computations can otherwise produce false membership claims.", "layers": [ { "id": "operation-boundaries", "name": "Operation boundaries", "description": "Distinguish factorial sums of digits from other arithmetic constructions.", "findings": [ { "id": "factorial-versus-power-and-place", "name": "Factorial versus power and place", "description": "Digit powers, factorials of whole integers, and factorial place-value expansions use different predicates and require neighbouring models.", "questions": [ { "text": "Is the expression a sum of digit factorials, rather than digit powers or a whole-number factorial?", "kind": "boundary", "id": "factorial-versus-power-and-place-q01" }, { "text": "Is a factorial numeral representation being mistaken for ordinary positional digits?", "kind": "boundary", "id": "factorial-versus-power-and-place-q02" } ] } ] }, { "id": "iteration-boundaries", "name": "Iteration boundaries", "description": "Separate immediate fixed points from eventual behaviour under repeated digit-factorial summation.", "findings": [ { "id": "fixed-point-versus-chain-member", "name": "Fixed point versus chain member", "description": "Membership requires S_b(n) = n after one application; reaching another fixed point or returning after a cycle of length greater than one is insufficient.", "questions": [ { "text": "Does the first application return the original candidate exactly?", "kind": "measurement", "id": "fixed-point-versus-chain-member-q01" }, { "text": "Should evidence about later iterates be linked to a digit-factorial-chain model instead?", "kind": "boundary", "id": "fixed-point-versus-chain-member-q02" } ] } ] } ] }, { "id": "evidence-and-external-alignment", "name": "Evidence and external alignment", "description": "What the world already says about this thing, gathered so the model can be checked against it.", "rationale": "A model that cannot be lined up against existing standards, identifiers and practice cannot be adopted by anyone who already uses them.", "layers": [ { "id": "reported-evidence", "name": "Reported evidence", "description": "Findings from the breadth pass, kept separate from the structural claims.", "findings": [ { "id": "evidence-confidence-notes", "name": "Check these first", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "The complete base-ten list is 1, 2, 145 and 40585; for example, 145 = 1! + 4! + 5!.", "Despite the batch's quantity framing, this is a number-theoretic property, not a physical quantity or measurement unit.", "This account is recalled knowledge; no sources were consulted." ], "questions": [ { "id": "evidence-confidence-notes-q01", "text": "Which of these check these first hold for the sense of factorion this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-real-world-use", "name": "Real-world use", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Examples in recreational number theory.", "Programming exercises involving digit extraction, factorials and exhaustive search." ], "questions": [ { "id": "evidence-real-world-use-q01", "text": "Which of these real-world use hold for the sense of factorion this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-failure-modes-and-hazards", "name": "Failure modes and hazards", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ "Omitting the numeral base: whether an integer is a factorion depends on its digit representation.", "Treating 0! as zero instead of one.", "Confusing the sum of digit factorials with the factorial of the digit sum." ], "questions": [ { "id": "evidence-failure-modes-and-hazards-q01", "text": "Which of these failure modes and hazards hold for the sense of factorion this model covers, and on what evidence?", "kind": "provenance" } ] }, { "id": "evidence-neighbours", "name": "Neighbouring kinds and how to tell them apart", "description": "Recalled without web access and unsourced; every item is a lead to verify.", "evidence": [ { "name": "factorial", "difference": "A factorial is the product n!; a factorion satisfies an equality involving the sum of factorials of its digits." }, { "name": "narcissistic number", "difference": "A narcissistic number equals the sum of its digits raised to the number of digits, whereas a factorion uses digit factorials." }, { "name": "factorial number system", "difference": "The factorial number system uses factorial place weights to represent integers; being a factorion is a property tested on an integer's digits in a specified positional base." } ], "questions": [ { "id": "evidence-neighbours-q01", "text": "Which of these neighbouring kinds and how to tell them apart hold for the sense of factorion this model covers, and on what evidence?", "kind": "provenance" } ] } ] } ] } ] }, "openQuestions": [ "Should the registry definition explicitly restrict unqualified factorion to decimal notation, or adopt the base-parameterised definition with decimal as a default?", "Which directly inspected proof or reproducible exhaustive computation should support the publication's claim that the decimal list is complete?", "Which alternative-base results warrant inclusion, and what evidence establishes their completeness?", "Which alternative names are documented as exact synonyms, and do any exclude the single-digit members?", "Does an existing Vercy world model already own this number class, requiring this registry entry to link to it?" ], "statistics": { "bundles": 6, "layers": 11, "findings": 14, "questions": 24 } }