math.special

Three distinct suites: educational examples on the calculator page, this legacy regression count (57/58 golden + boundary), and the CVP verification suite below. They are not the same number. Property, metamorphic, round-trip, O3, and interface statuses/checks are recorded in the CVP Manifest . SKIP (0/0) means that suite has no published vectors.

This calculator CURRENT · Public schema 1.10.6 matches · Semantic contract ✓ · Production attested · Public/cache ✓ · Origin ✓

CVP verification suite

Protocol checks from the Evidence Manifest. Distinct from the legacy regression suite above. Verification revision 2026-10-19.special-full-review-v8. Calculation version 1.10.6. CVP 1.0.0-proposed. 1000/1000 golden · 8/8 CVP boundary · 24/24 invalid · 1/1 property · 3/3 metamorphic · 24/24 cross-interface · 1/1 cross-calculator · 6/6 CVP contract. digest a3f6d148d541.

• Golden (oracle-backed) 1000 / 1000
• CVP boundary 8 / 8
• Invalid input 24 / 24
• Property 1 / 1
• Metamorphic 3 / 3
• Cross-interface 24 / 24
• Cross-calculator 1 / 1
• CVP contract 6 / 6
• Interfaces PASS · UI (SSR) / REST / MCP
• Interface claim ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only. Special response contract also checks all 14 mode formulas and unrounded numeric summaries, including erfc/K tails and a subnormal erf input; UI is shared presentation/SSR, not a live browser session.
• Verified scope all published modes
• Assurance Core

• release artifacts aligned PASS
• schema output required declared PASS
• schema valid input executes PASS
• engine success output validates schema PASS
• schema rejects incomplete input PASS
• rest success payload validates schema PASS

Fingerprint sha256:5fbfcc9d018c864dfb99283b1b2d9d11a5e1174a08368648093fec31af92a715

Input schema sha256:edc3a9ac82fb12bade39bb5baa124a9e31d9b5289f56b19ca7c02500463344dd · Output schema sha256:5864981f912b9805e21ea9ddf0882a27e408999a948a3b5b94bbe1de8b76fe9c · 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.

Tool

special-functions

Engine

special-functions

Engine build

2026-10-19

Evidence core

2026-10-19T04:04:19.258Z

Rendered

2026-10-23T01:27:11.644Z

Last reviewed

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Assumptions

• Relative accuracy weakens near zeros of Bessel J/Y. Small positive K tails are retained within Float64 range.
• 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
• The Result Card includes an engine-linked Live graph for erf, erfc, Γ, ln Γ, J_n / Y_n / I_n / K_n, W₀ / W₋₁, B(x, y) vs x (y held), incomplete B_x on [0, 1], sinc, and normalized sinc. Same special-functions engine as REST/MCP. Integer n! stays on /factorial. Not CAS.

Limitations

• Real, dimensionless binary64 inputs; publishes12 significant digits, no arbitrary-precision/exact claim.
• Default mode erf and Bessel order0 apply only when absent; explicit empty/null rejects. All supplied numeric aliases must agree; known irrelevant numeric parameters are parsed then ignored.
• Gamma poles reject. Nonzero Gamma that underflows rejects NUMERICAL_RANGE_ERROR; positive overflow suggests lgamma. lgamma is computed directly for x>0 with a stable Taylor expansion near1 and2.
• Bessel integer n0–8, |x|≤80, Y/K require x>0; subnormal error and zeros limit relative accuracy.
• Lambert W0 and W-1 explicit real branches; binary64 -1/e designated endpoint, no tolerance band absorbs adjacent inputs.
• Beta shapes positive and at most1e6, except exact B(x,1)=1/x or B(1,y)=1/y. Incomplete beta shapes at most1e6. CF capped10000 steps; failure NO_CONVERGENCE.
• erf/erfc and beta tails can round to zero only on binary64 underflow, not an absolute display cutoff. Regularized incomplete beta evaluated separately in logs.
• Normalized sinc uses sinpi reduction and returns zero at representable nonzero integer x. Other tiny nonzero results retain significance.
• Numeric references have relative1e-10, with J/Y near zeros absolute2e-14,4 minimum subnormals otherwise. No global error proof for all real inputs.
• Browser interaction computes locally; parameterized share pages, REST and MCP send inputs to server. Logs/retention/region are deployment concerns; no zero-log promise.