{
  "capability_id": "math.factorial",
  "tool_id": "factorial",
  "engine": "factorial",
  "calculation_version": "1.0.2",
  "generated_at": "2026-09-16T22:03:43.768Z",
  "engine_build": "2026-09-10",
  "test_count": 24,
  "passed": 24,
  "failed": 0,
  "tests_summary": {
    "golden": {
      "total": 8,
      "passed": 8
    },
    "boundary": {
      "total": 7,
      "passed": 7
    },
    "property": {
      "total": 9,
      "passed": 9
    }
  },
  "engine_tested": true,
  "expert_reviewed": false,
  "internally_reviewed": false,
  "review_kind": null,
  "source_checked": true,
  "reviewed_by": null,
  "reviewer_role": null,
  "last_reviewed": null,
  "expert_review": {
    "status": "optional_not_performed",
    "reviewer": null,
    "scope": [
      "formula",
      "input semantics",
      "assumptions",
      "limitations",
      "sources",
      "tests"
    ],
    "note": "Domain expert review is optional under CVP (CVP-EXP-01). Absence does not block CVP Verified. See Review policy for supplemental domain assurance."
  },
  "internal_review": {
    "status": "not_applicable",
    "reviewer": null
  },
  "page": "https://www.calculatorx.com/calc/math/factorial",
  "api": "https://www.calculatorx.com/api/v1/calc/factorial",
  "schema_input": "https://www.calculatorx.com/schemas/math.factorial.input.json",
  "schema_output": "https://www.calculatorx.com/schemas/math.factorial.output.json",
  "assumptions": [
    "n is an integer satisfying 0 ≤ n ≤ 5000.",
    "Ordinary (single) factorial only — not multifactorial and not the real Gamma function for non-integers."
  ],
  "limitations": [
    "Ordinary (single) factorial only — not multifactorial and not Γ(n+1) for non-integers",
    "Domain is 0 ≤ n ≤ 5000 (product/resource bound, not a mathematical bound)",
    "Missing n → MISSING_REQUIRED_INPUT",
    "Non-numeric or non-integer n → INVALID_NUMBER",
    "n < 0 → VALUE_MUST_BE_NON_NEGATIVE",
    "n > 5000 → VALUE_ABOVE_MAX"
  ],
  "boundary_conditions": [
    "Missing n → MISSING_REQUIRED_INPUT",
    "Non-numeric n → INVALID_NUMBER",
    "Non-integer n (for example 5.5) → INVALID_NUMBER",
    "n < 0 → VALUE_MUST_BE_NON_NEGATIVE",
    "n > 5000 → VALUE_ABOVE_MAX"
  ],
  "sources": [
    {
      "title": "NIST Digital Library of Mathematical Functions, Chapter 5",
      "section": "Gamma Function",
      "url": "https://dlmf.nist.gov/5",
      "supports": "Ordinary factorial of a nonnegative integer; n! = Γ(n+1)",
      "kind": "reference"
    },
    {
      "title": "NIST DLMF §5.4",
      "section": "Special values and extrema",
      "url": "https://dlmf.nist.gov/5.4",
      "supports": "n! = Γ(n+1), including 0! = 1. This calculator publishes ordinary factorial only; non-integer Gamma values are out of scope.",
      "kind": "reference"
    },
    {
      "title": "CalculatorX mathematical conventions",
      "section": "Domain 0 ≤ n ≤ 5000; exact BigInt n!",
      "supports": "Matches the on-page contract. Permutation count P(n,n)=n! is an interpretation, not a second engine.",
      "kind": "definition"
    }
  ],
  "methods": [],
  "fingerprint": "sha256:316d01876698831ff0c9b1d2d351d3c8c971e92e92f1898ee98f855297b07420",
  "tests": [
    {
      "id": "zero",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "n": 0
      },
      "expect": {
        "value": 1,
        "digits": 1
      },
      "expectWarning": null,
      "actual": {
        "n": 0,
        "value": 1,
        "exact": "1",
        "digits": 1,
        "expansion": "0! = 1  (by definition)",
        "scientific": "1 × 10^0",
        "max_n": 5000
      },
      "tol": 0
    },
    {
      "id": "one",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "n": 1
      },
      "expect": {
        "value": 1,
        "digits": 1
      },
      "expectWarning": null,
      "actual": {
        "n": 1,
        "value": 1,
        "exact": "1",
        "digits": 1,
        "expansion": "1! = 1",
        "scientific": "1 × 10^0",
        "max_n": 5000
      },
      "tol": 0
    },
    {
      "id": "five",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "n": 5
      },
      "expect": {
        "value": 120,
        "digits": 3
      },
      "expectWarning": null,
      "actual": {
        "n": 5,
        "value": 120,
        "exact": "120",
        "digits": 3,
        "expansion": "5! = 5 × 4 × 3 × 2 × 1",
        "scientific": "1.2 × 10^2",
        "max_n": 5000
      },
      "tol": 0
    },
    {
      "id": "ten",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "n": 10
      },
      "expect": {
        "value": 3628800,
        "digits": 7
      },
      "expectWarning": null,
      "actual": {
        "n": 10,
        "value": 3628800,
        "exact": "3628800",
        "digits": 7,
        "expansion": "10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1",
        "scientific": "3.6288 × 10^6",
        "max_n": 5000
      },
      "tol": 0
    },
    {
      "id": "twenty",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "n": 20
      },
      "expect": {
        "value": "2432902008176640000",
        "digits": 19
      },
      "expectWarning": null,
      "actual": {
        "n": 20,
        "value": "2432902008176640000",
        "exact": "2432902008176640000",
        "digits": 19,
        "expansion": "20! = 20 × 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1",
        "scientific": "2.43290200817 × 10^18",
        "max_n": 5000
      },
      "tol": 0
    },
    {
      "id": "hundred",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "n": 100
      },
      "expect": {
        "digits": 158
      },
      "expectWarning": null,
      "actual": {
        "n": 100,
        "value": "93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000",
        "exact": "93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000",
        "digits": 158,
        "expansion": "100! = 100 × (99)!",
        "scientific": "9.33262154439 × 10^157",
        "max_n": 5000
      },
      "tol": 0
    },
    {
      "id": "thousand",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "n": 1000
      },
      "expect": {
        "digits": 2568
      },
      "expectWarning": null,
      "actual": {
        "n": 1000,
        "value": "402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "exact": "402387260077093773543702433923003985719374864210714632543799910429938512398629020592044208486969404800479988610197196058631666872994808558901323829669944590997424504087073759918823627727188732519779505950995276120874975462497043601418278094646496291056393887437886487337119181045825783647849977012476632889835955735432513185323958463075557409114262417474349347553428646576611667797396668820291207379143853719588249808126867838374559731746136085379534524221586593201928090878297308431392844403281231558611036976801357304216168747609675871348312025478589320767169132448426236131412508780208000261683151027341827977704784635868170164365024153691398281264810213092761244896359928705114964975419909342221566832572080821333186116811553615836546984046708975602900950537616475847728421889679646244945160765353408198901385442487984959953319101723355556602139450399736280750137837615307127761926849034352625200015888535147331611702103968175921510907788019393178114194545257223865541461062892187960223838971476088506276862967146674697562911234082439208160153780889893964518263243671616762179168909779911903754031274622289988005195444414282012187361745992642956581746628302955570299024324153181617210465832036786906117260158783520751516284225540265170483304226143974286933061690897968482590125458327168226458066526769958652682272807075781391858178889652208164348344825993266043367660176999612831860788386150279465955131156552036093988180612138558600301435694527224206344631797460594682573103790084024432438465657245014402821885252470935190620929023136493273497565513958720559654228749774011413346962715422845862377387538230483865688976461927383814900140767310446640259899490222221765904339901886018566526485061799702356193897017860040811889729918311021171229845901641921068884387121855646124960798722908519296819372388642614839657382291123125024186649353143970137428531926649875337218940694281434118520158014123344828015051399694290153483077644569099073152433278288269864602789864321139083506217095002597389863554277196742822248757586765752344220207573630569498825087968928162753848863396909959826280956121450994871701244516461260379029309120889086942028510640182154399457156805941872748998094254742173582401063677404595741785160829230135358081840096996372524230560855903700624271243416909004153690105933983835777939410970027753472000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "digits": 2568,
        "expansion": "1000! = 1000 × (999)!",
        "scientific": "4.02387260077 × 10^2567",
        "max_n": 5000
      },
      "tol": 0
    },
    {
      "id": "max",
      "ok": true,
      "kind": "golden",
      "inputs": {
        "n": 5000
      },
      "expect": {
        "digits": 16326
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      "expectWarning": null,
      "actual": {
        "n": 5000,
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"4228577926605543522201064200233584405390786674626646748849782402181358052708108200690899047871706387537084746657300685445878486066683812736337210893772787631279390363058462160643904478986982239871929708896211612652968321775500399242196837031469072644728787897904047548841622152266719284109692369104495659717363529484002238403811206448202308576711045023061748947554283097617817240408053248099278093287840554861993645482912118762582488021891739779000502132125980436392446264607705113588465951086754705858339246552255890354744359883473831789880346330084586315102090915099356538200109330479657425567419309170551728052002360750859911976352287559079020433697431235069168312119244959715562674075214621989862330886259983028598648575787494459631152869708867100462684236481789899054546908613916132183441741488071862344481148312094903611965468727677556178868287202691048140924564103418359756042764581615131785759016610717825441569808833593727299956033713712004710494376562911424886053352994996423006999722049181201008190594391406750532650047755338508990979451015510914869070044071195723360262433681323302187092876991968066565697527904222582678415610833764257810326292026872110702746813943511286015023261906499591718973641763784364912197091098409445148953589591038041769419566578348220717491055127526391483811720526048269651626427100949193933326610301043605304591170145572095847143537219482466867934673759048722681334102078609036571088063766162497495074131070774016821805859455264451714092774692300626975113460441745679467358287822616295842486751573791729427241787831054298582451175755118845065744248275746608002385883784923962473687615070157677258983211286322955370449025163879251275908417917446404669135310473479844649961545955420139963173574763017400367961929199421907628954456562617670417995381611333873128235115341525813090879158836383516647972259129442706535571425117373238072326329581217979166796923296870969239010032555747890550998074870610472306461959849552396576122086738665141716993075576918979026751573420758647963453384468350859654907273263219105040642897130962245051620646694680988699171221274045040206849232662417601329102278666872703052847094525268254966177724996452066998369259106908940826374010434983715911264558222806063613941153443167717699343536642849282944364147696158819936613882555774877099370045947539078451490344345211745605940399162684446976618213874707053255595779331964609966621453775649354741697085623892147732228655071824904300161861421927604523076706211429617672747041236161072200097437586474927536651495321647808490751463300710166913134206628825626182838658369836321087607104275160733483477884147967324270804108607618412818883071150989821353384066106521470870468747609954274736735094515535997690403673533855510525716826503176824057439934148623923319814325791821933218989404508650136109980983839931109963559813280010497315885963121318538012050467876429106693656004373056334319848790489985247012933007893445328681566797628804955328463860201334802652798369463933849956750499937078147465615434389304313842378789818478028860099710886956329883477118631223827859636531151323779313736473974293694114990287519722279995451826154882989511519266821124513553184722099904353559498872999220350620398160110863762365397821723802378466506736245106350344231873153383082120438047109994192278210397475527174160438901697239613055493718448361198035658960620250090936643993601720073836133544050943290724765189095025077246758419894122246593921631163520381473624795285397320893095334219106357028055766297201565565107677808059334536311218295617928876730028024509321227788529684182082617784769556449803856912757873726780409591587117339711031652326780607981276092461735041201826668742628053852758439791676090077433807484207511851191029219603393762809867536650852128692553215367879325218825741018661370543289737358627253701785588066398513503869440396049282588201804191780736496938858025977583988920143897471654659735108526057062344020696370656601295357340435829614734272758056308395106673753492596595185756469397232182757800032505938953038205396975588705115430739208274224405162997087395997684612062466290981123680125798912848025050940289169597650793954371913113793144274051355996303756422145272943417972461875979640742391478389935415658347161568584990367730566113538333670875489004130919816763307490415103375973072468858392469417155482957307506185058815819595289926602256269034395733134506669729521152306686962279209477799743365744726734714089280714112838880826933773780772931041107675136394762006108580405960196390580157610023374638693522283858014349571781255814458629300424794040657368598620079146045902554139299500880447103847589903265480973381669405000854527237135713949024638203086685418028383175276680642784895610057558599917189667864491540635700144971942498789208597312542755675145752063991181507363974831024907938417256534218942767691165981534300846370877695102954151365517346750540152397060425717460010899684404988459854779779050316325684891565572310064997264987214808001817703577015029830088794872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2371367395995678940884705976859514505017241517746017351430990972615509378334720000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "digits": 16326,
        "expansion": "5000! = 5000 × (4999)!",
        "scientific": "4.2285779266 × 10^16325",
        "max_n": 5000
      },
      "tol": 0
    },
    {
      "id": "negative",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "n": -1
      },
      "expectError": "VALUE_MUST_BE_NON_NEGATIVE",
      "code": "VALUE_MUST_BE_NON_NEGATIVE",
      "error": "n must be ≥ 0."
    },
    {
      "id": "fraction",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "n": 5.5
      },
      "expectError": "INVALID_NUMBER",
      "code": "INVALID_NUMBER",
      "error": "n must be a non-negative whole number (digits only)."
    },
    {
      "id": "above-max",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "n": 5001
      },
      "expectError": "VALUE_ABOVE_MAX",
      "code": "VALUE_ABOVE_MAX",
      "error": "n must be at most 5000 for this calculator."
    },
    {
      "id": "empty",
      "ok": true,
      "kind": "boundary",
      "inputs": {},
      "expectError": "MISSING_REQUIRED_INPUT",
      "code": "MISSING_REQUIRED_INPUT",
      "error": "Enter a non-negative integer n."
    },
    {
      "id": "boundary-alias-nInput",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "nInput": 5
      },
      "expect": {
        "n": 5,
        "value": 120,
        "digits": 3
      },
      "expectWarning": null,
      "actual": {
        "n": 5,
        "value": 120,
        "exact": "120",
        "digits": 3,
        "expansion": "5! = 5 × 4 × 3 × 2 × 1",
        "scientific": "1.2 × 10^2",
        "max_n": 5000
      },
      "tol": null
    },
    {
      "id": "boundary-non-numeric",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "n": "abc"
      },
      "expectError": "INVALID_NUMBER",
      "code": "INVALID_NUMBER",
      "error": "n must be a non-negative whole number (digits only)."
    },
    {
      "id": "boundary-half-integer",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "n": 1.5
      },
      "expectError": "INVALID_NUMBER",
      "code": "INVALID_NUMBER",
      "error": "n must be a non-negative whole number (digits only)."
    },
    {
      "id": "property-empty-product",
      "ok": true,
      "kind": "property",
      "detail": "0! = 1",
      "expect": 1,
      "actual": 1
    },
    {
      "id": "property-recurrence",
      "ok": true,
      "kind": "property",
      "detail": "n! = n×(n−1)! for n = 2..40",
      "expect": 39,
      "actual": 39
    },
    {
      "id": "property-quotient",
      "ok": true,
      "kind": "property",
      "detail": "n!/(n−1)! = n for n = 2..40",
      "expect": 39,
      "actual": 39
    },
    {
      "id": "property-divisible-by-n",
      "ok": true,
      "kind": "property",
      "detail": "n! mod n = 0 for n = 2..40",
      "expect": 39,
      "actual": 39
    },
    {
      "id": "property-recurrence-sparse",
      "ok": true,
      "kind": "property",
      "detail": "n! = n×(n−1)! for n ∈ {47, 61, 73, 89, 100}",
      "expect": 5,
      "actual": 5
    },
    {
      "id": "property-digits-log10",
      "ok": true,
      "kind": "property",
      "detail": "digits(n!) = ⌊Σ log10(k)⌋ + 1",
      "expect": 8,
      "actual": 8
    },
    {
      "id": "property-deterministic",
      "ok": true,
      "kind": "property",
      "detail": "20! is deterministic",
      "expect": "2432902008176640000",
      "actual": "2432902008176640000"
    },
    {
      "id": "property-alias-nInput",
      "ok": true,
      "kind": "property",
      "detail": "nInput aliases n",
      "expect": 120,
      "actual": 120
    },
    {
      "id": "property-max-n-bound",
      "ok": true,
      "kind": "property",
      "detail": "max_n = 5000",
      "expect": 5000,
      "actual": 5000
    }
  ],
  "tier": "engine_tested",
  "verification": {
    "tests": {
      "passed": 24,
      "total": 24,
      "golden": 8,
      "boundary": 7,
      "property": 9
    },
    "contract_checks": {
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      "items": [
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        {
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      ]
    },
    "report": "https://www.calculatorx.com/.well-known/calculatorx-release-integrity.json"
  }
}