Public evidence

math.special

Machine-checked verification for calculation version 1.10.0 · tier engine_tested

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Verification

58/58 calculation tests · Production surface contract 4/4

Golden tests 13 / 13
Boundary tests 8 / 8
Property tests 37 / 37
Production surface contract 4 / 4
Engine tested Yes
Source checked Yes
Artifact integrity PASS
Production attestation Embedded snapshot: unpublished · Build schema 1.10.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (2 capabilities; 162 remain CURRENT) @ 2026-09-20T09:01:12.147Z
Semantic contract PASS
Evidence attested at 2026-09-18T21:41:55.460Z
Last production check 2026-09-18T21:41:55.460Z
Public / cache —
Origin —
Supplemental domain review Not performed
Named expert review Not performed

release artifacts aligned PASS
schema valid input executes PASS
engine success output validates schema PASS
schema rejects incomplete input PASS

Fingerprint sha256:51533eecfeaa78ed9a19aecf37760944d857b430aa676bef46b8f89b684f96af

Input schema sha256:8b9159f42671dafa4cf80275c34395c83fae0f5d0be7dcc2342c329578a66f01 · Output schema sha256:4240a4b6d19a781570dec236229987f2533d42c9fcadede16a0d5c3c3112d3d9 · Build-time SHA-256 digests prove published artifacts agree with each other (release_integrity.scope=workspace). Production attestation GETs canonical URLs twice (public/cached view and origin), including Capability HTML, via npm run attest:production --write (published at /.well-known/calculatorx-production-attestation.json) and fails if the public Capability page advertises a different calculation_version. This is artifact consistency, not an independent immutable release log.

Build & scope

Engine identity and declared limitations.

Tool: special-functions
Engine: special-functions
Engine build: 2026-09-20
Evidence core: 2026-09-20T07:29:05.922Z
Rendered: 2026-09-20T09:07:29.782Z
Last reviewed: —

Assumptions

Real argument
Gamma poles at non-positive integers
lgamma is real for x > 0
Bessel J integer n=0…8, |x|≤80
Bessel Y integer n=0…8, x>0
Bessel I integer n=0…8, |x|≤80
Bessel K integer n=0…8, x>0
Lambert W_0 for x ≥ −1/e
Lambert W_{-1} for −1/e ≤ x < 0
Complete Euler B(x,y) for x>0, y>0
Incomplete B_x(a,b) for a>0, b>0, 0≤x≤1
Unnormalized sinc(x)=sin(x)/x, sinc(0)=1
Normalized sinc(x)=sin(πx)/(πx), sinc(0)=1
Not factorial
Not CAS

Limitations

Real argument only
Not factorial — integer n! stays on math.factorial
Gamma poles at non-positive integers
lgamma is real for x > 0 on this seed
Bessel J / Y / I / K integer n=0…8, |x|≤80 for J and I, x>0 for Y and K
Lambert W_0 for x ≥ −1/e; W_{-1} for −1/e ≤ x < 0
Complete Euler B(x,y) for x>0, y>0; incomplete B_x(a,b) for a>0, b>0, 0≤x≤1
Unnormalized sinc(x)=sin(x)/x and normalized sinc(x)=sin(πx)/(πx), both with sinc(0)=1
unknown mode (including cas, factorial) → INVALID_MODE

Sources

Standards and references supporting this tool specification. Methods are listed separately — they are algorithms, not bibliographic sources.

ISO 80000-2:2019 Quantities and units — Mathematics

Published checks

Golden, boundary, and property cases included in this evidence build.

ID	Kind	Status	Detail
`golden-erf-1`	golden	pass	Expected value=0.84270079295 · Actual value=0.84270079295 · mode=erf · x=1 · erf=0.84270079295 · erfc=0.15729920705 · formula=erf(x) = (2/√π) ∫₀ˣ e^{−t²} dt
`golden-gamma-half`	golden	pass	Expected value=1.77245385091 · Actual value=1.77245385091 · mode=gamma · x=0.5 · gamma=1.77245385091 · formula=Γ(x) = ∫₀^∞ t^{x−1} e^{−t} dt
`golden-gamma-6`	golden	pass	Expected value=120 · Actual value=120 · mode=gamma · x=6 · gamma=120 · formula=Γ(x) = ∫₀^∞ t^{x−1} e^{−t} dt
`golden-j0-1`	golden	pass	Expected n=0 · value=0.765197686558 · algorithm=series · Actual n=0 · value=0.765197686558 · mode=bessel_j · x=1 · formula=J_n(x) = (x/2)^n Σ_k (−1)^k (x/2)^{2k} / (k! (n+k)!) · algorithm=series
`golden-w-e`	golden	pass	Expected value=1 · branch=0 · algorithm=halley · Actual value=1 · mode=lambert_w · x=2.718281828459045 · branch=0 · formula=W(x) e^{W(x)} = x · algorithm=halley
`golden-b23`	golden	pass	Expected value=0.0833333333333 · algorithm=lgamma · Actual value=0.0833333333333 · mode=beta · x=2 · y=3 · formula=B(x, y) = Γ(x)Γ(y) / Γ(x+y) · algorithm=lgamma
`golden-sinc-0`	golden	pass	Expected value=1 · algorithm=unnormalized · Actual value=1 · mode=sinc · x=0 · formula=sinc(x) = sin(x)/x, sinc(0)=1 · algorithm=unnormalized
`golden-y0-1`	golden	pass	Expected n=0 · value=0.0882569642157 · algorithm=series · Actual n=0 · value=0.0882569642157 · mode=bessel_y · x=1 · formula=Y_n(x) = (2/π) J_n(x) ln(x/2) − (x/2)^{−n}/π Σ_{k<n} − (x/2)^n/π Σ ψ · algorithm=series
`golden-i0-1`	golden	pass	Expected n=0 · value=1.26606587775 · algorithm=series · Actual n=0 · value=1.26606587775 · mode=bessel_i · x=1 · formula=I_n(x) = (x/2)^n Σ_k (x/2)^{2k} / (k! (n+k)!) · algorithm=series
`golden-k0-1`	golden	pass	Expected n=0 · value=0.421024438241 · algorithm=series · Actual n=0 · value=0.421024438241 · mode=bessel_k · x=1 · formula=K_n(x) = ½ (x/2)^{−n} Σ_{k<n} (−1)^k + (−1)^{n+1} I_n ln(x/2) + (−1)^n ½ (x/2)^n Σ ψ · algorithm=series
`golden-wm1-01`	golden	pass	Expected value=-3.57715206396 · branch=-1 · algorithm=halley · Actual value=-3.57715206396 · mode=lambert_wm1 · x=-0.1 · branch=-1 · formula=W_{-1}(x) e^{W_{-1}(x)} = x · algorithm=halley
`golden-ibeta-half`	golden	pass	Expected value=0.0572916666667 · regularized=0.6875 · algorithm=series · Actual value=0.0572916666667 · mode=incomplete_beta · x=0.5 · a=2 · b=3 · regularized=0.6875 · formula=B_x(a, b) = ∫₀ˣ t^{a−1} (1−t)^{b−1} dt · algorithm=series
`golden-nsinc-half`	golden	pass	Expected value=0.636619772368 · algorithm=normalized · Actual value=0.636619772368 · mode=normalized_sinc · x=0.5 · formula=sinc(x) = sin(πx)/(πx), sinc(0)=1 · algorithm=normalized
`boundary-special-mode`	boundary	pass	Expected INVALID_MODE · Actual INVALID_MODE
`boundary-gamma-pole`	boundary	pass	Expected INVALID_INPUT · Actual INVALID_INPUT
`boundary-y0`	boundary	pass	Expected INVALID_INPUT · Actual INVALID_INPUT
`boundary-i0`	boundary	pass	Expected INVALID_INPUT · Actual INVALID_INPUT
`boundary-k0`	boundary	pass	Expected INVALID_INPUT · Actual INVALID_INPUT
`boundary-wm1`	boundary	pass	Expected INVALID_INPUT · Actual INVALID_INPUT
`boundary-ibeta`	boundary	pass	Expected INVALID_INPUT · Actual INVALID_INPUT
`boundary-nsinc`	boundary	pass	Expected INVALID_NUMBER · Actual INVALID_NUMBER
`property-erf-1`	property	pass	erf(1)
`property-erf-odd`	property	pass	erf odd
`property-erf-erfc`	property	pass	erf+erfc
`property-gamma-6`	property	pass	Γ(6)=5!
`property-gamma-half`	property	pass	Γ(1/2)=√π
`property-lgamma-6`	property	pass	ln Γ(6)
`property-j0-0`	property	pass	J_0(0)=1
`property-j1-1`	property	pass	J_1(1)
`property-j1-odd`	property	pass	J_1 odd
`property-w0-0`	property	pass	W_0(0)=0
`property-w-e`	property	pass	W_0(e)=1
`property-w-identity`	property	pass	W e^W = x
`property-beta-11`	property	pass	B(1,1)=1
`property-beta-half`	property	pass	B(1/2,1/2)=π
`property-beta-sym`	property	pass	B(x,y)=B(y,x)
`property-sinc-0`	property	pass	sinc(0)=1
`property-sinc-pi`	property	pass	sinc(π)=0
`property-sinc-even`	property	pass	sinc even
`property-y0-1`	property	pass	Y_0(1)
`property-y1-1`	property	pass	Y_1(1)
`property-i0-0`	property	pass	I_0(0)=1
`property-i0-1`	property	pass	I_0(1)
`property-i1-odd`	property	pass	I_1 odd
`property-k0-1`	property	pass	K_0(1)
`property-k-recurrence`	property	pass	K_2 = (2/x)K_1 + K_0
`property-wm1-branch`	property	pass	W_{-1}(-1/e)=-1
`property-wm1-m2`	property	pass	W_{-1}(-2/e²)=-2
`property-wm1-identity`	property	pass	W_{-1} e^{W}=x
`property-ibeta-0`	property	pass	B_0(a,b)=0
`property-ibeta-1`	property	pass	B_1(a,b)=B(a,b)
`property-ibeta-11`	property	pass	B_x(1,1)=x
`property-ibeta-half`	property	pass	B_{1/2}(1/2,1/2)=π/2
`property-ibeta-identity`	property	pass	B_x(a,b)+B_{1-x}(b,a)=B
`property-nsinc-0`	property	pass	sinc_N(0)=1
`property-nsinc-1`	property	pass	sinc_N(1)=0
`property-nsinc-even`	property	pass	sinc_N even
`property-nsinc-identity`	property	pass	sinc_N(1/2)=sinc(π/2)