{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.factorion","code":"thing-q1977759","url":"https://ver.cy/models/thing/q1977759/","name":"factorion","alternateNames":[],"kind":"thing","status":"published","version":"0.1.0-research.1","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A factorion is a positive integer equal to the sum of the factorials of its decimal digits.","purpose":"Enable an agent to recognise a factorion, verify its decimal digit-factorial equality, and judge whether a claimed enumeration is complete.","scope":{"in":["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":["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"],"boundaries":[]},"distinguishingFeatures":["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."],"structure":{"bundles":[{"id":"factorion-membership","name":"Factorion membership","description":"The defining equality and the numerical domain in which it applies.","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?","id":"digit-factorial-fixed-point-q01","kind":"definition"},{"text":"Does the claim apply factorial to each digit separately before summing?","id":"digit-factorial-fixed-point-q02","kind":"boundary"}]}]},{"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?","id":"integer-domain-and-zero-q01","kind":"boundary"},{"text":"Does the verifier correctly reject zero, negative values, and nonintegers as members?","id":"integer-domain-and-zero-q02","kind":"action"}]}]}]},{"id":"base-and-digit-semantics","name":"Base and digit semantics","description":"The representation choices that determine the factorial sum.","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?","id":"base-qualified-membership-q01","kind":"definition"},{"text":"When changing base, has the agent preserved the integer value and recomputed its digits?","id":"base-qualified-membership-q02","kind":"action"}]}]},{"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?","id":"zero-and-multiplicity-q01","kind":"measurement"},{"text":"Are repeated digits counted separately and all genuine zero digits assigned factorial value 1?","id":"zero-and-multiplicity-q02","kind":"measurement"}]}]}]},{"id":"exact-membership-verification","name":"Exact membership verification","description":"Evidence sufficient to accept or reject an individual candidate.","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?","id":"recomputable-equality-q01","kind":"measurement"},{"text":"Can an independent calculation reproduce the zero or nonzero residual without rounding or overflow?","id":"recomputable-equality-q02","kind":"action"}]}]},{"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?","id":"known-decimal-members-q01","kind":"action"},{"text":"Which exact calculation supports each accepted reference case and each chosen nonmember test?","id":"known-decimal-members-q02","kind":"measurement"}]}]}]},{"id":"finite-enumeration","name":"Finite enumeration","description":"Search limits, candidate generation, and evidence that no members were omitted.","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?","id":"necessary-size-inequality-q01","kind":"measurement"},{"text":"What argument establishes that every length beyond the chosen cutoff also fails the inequality?","id":"necessary-size-inequality-q02","kind":"definition"}]}]},{"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?","id":"coverage-and-reconstruction-q01","kind":"provenance"},{"text":"For a digit-multiset search, does each retained sum reproduce the same base-b digit multiplicities?","id":"coverage-and-reconstruction-q02","kind":"action"}]}]}]},{"id":"neighbouring-number-classes","name":"Neighbouring number classes","description":"Boundaries against similar factorial and digit-based constructions.","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?","id":"factorial-versus-power-and-place-q01","kind":"boundary"},{"text":"Is a factorial numeral representation being mistaken for ordinary positional digits?","id":"factorial-versus-power-and-place-q02","kind":"boundary"}]}]},{"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?","id":"fixed-point-versus-chain-member-q01","kind":"measurement"},{"text":"Should evidence about later iterates be linked to a digit-factorial-chain model instead?","id":"fixed-point-versus-chain-member-q02","kind":"boundary"}]}]}]},{"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.","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.","questions":[{"text":"Which of these check these first hold for the sense of factorion this model covers, and on what evidence?","id":"evidence-confidence-notes-q01","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.","questions":[{"text":"Which of these real-world use hold for the sense of factorion this model covers, and on what evidence?","id":"evidence-real-world-use-q01","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.","questions":[{"text":"Which of these failure modes and hazards hold for the sense of factorion this model covers, and on what evidence?","id":"evidence-failure-modes-and-hazards-q01","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.","questions":[{"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?","id":"evidence-neighbours-q01","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Find, verify and explain decimal factorions.","Explain why only finitely many decimal factorions exist."],"mustNot":["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."],"requiresHuman":["Publishing new results under someone else's name requires human involvement."]},"ethics":{"considerations":["Recreational mathematics invites learners in, so wrong claims mislead them."],"affectedParties":["Students","Puzzle enthusiasts"]},"owners":{"steward":"Nobody owns them; they are held in common as mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.factorion","Wikidata: Q1977759 (https://www.wikidata.org/wiki/Q1977759)"]},"properties":{"applicability":"not-applicable","items":["Candidate integer: Exact positive integer; no physical unit","Numeral base: Integer b >= 2; decimal convention b = 10","Canonical digit sequence: Digits in 0 through b - 1, with no leading zeros","Digit multiplicities: Count c_d >= 0 for each allowed digit d","Digit-factorial sum: Exact integer S_b(n) = sum(d!)","Equality residual: Exact integer S_b(n) - n","Verification state: Untested | verified member | verified nonmember | invalid input","Enumeration coverage: Examples only | bounded search completed | exhaustive enumeration justified"]},"recognition":{"applicability":"not-applicable","items":[]},"capabilities":{"applicability":"required","items":["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."]},"hazards":{"applicability":"required","items":["Errors from forgetting that 0! equals 1.","Mixing results across numeral bases."]},"interfaces":{"applicability":"not-applicable","items":[]},"context":{"applicability":"required","items":["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."]}},"sources":[{"title":"Wikidata item Q1977759: factorion","url":"https://www.wikidata.org/wiki/Q1977759","note":"identity and sense of the item"},{"title":"Wikipedia: Factorion","url":"https://en.wikipedia.org/wiki/Factorion","note":"general description of the item"}],"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?"],"resources":{"spec":"/models/things/publications/thing-q1977759/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1977759/spec.json"],"providers":["Codex"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T09:37:18Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"not-applicable","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"not-applicable","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-07T12:16:30+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties.","interaction.interfaces":"No specific standard, law or official classification governs factorions as mathematical objects.","interaction.recognition":"A factorion is an abstract integer property established by calculation rather than visual appearance."},"score":1.0}}