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    "Relative accuracy weakens near zeros of Bessel J/Y. Small positive K tails are retained within Float64 range.",
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    "Bessel J integer n=0…8, |x|≤80",
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    "Bessel I integer n=0…8, |x|≤80",
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    "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."
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    "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."
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    "non-finite x → INVALID_NUMBER",
    "Gamma at 0, −1, −2, … → INVALID_INPUT",
    "lgamma with x ≤ 0 → INVALID_INPUT",
    "Bessel n not in 0…8 or |x|>80 → INVALID_INPUT",
    "Y_n or K_n with x ≤ 0 → INVALID_INPUT",
    "Lambert W_0 x < −1/e → INVALID_INPUT",
    "Lambert W_{-1} x < −1/e or x ≥ 0 → INVALID_INPUT",
    "Beta x ≤ 0 or y ≤ 0 → INVALID_INPUT",
    "Incomplete beta a ≤ 0, b ≤ 0, or x ∉ [0,1] → INVALID_INPUT",
    "unknown mode → INVALID_MODE"
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      "title": "ISO 80000-2:2019 Quantities and units — Mathematics",
      "url": "https://www.iso.org/standard/64973.html",
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      "kind": "standard",
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      "title": "Abramowitz and Stegun, Handbook of Mathematical Functions — erf, Gamma, Bessel, incomplete beta",
      "supports": "Error function 7.1.26; Γ(n)=(n−1)!; integer-order J_n/Y_n/I_n/K_n; incomplete beta integral",
      "kind": "reference",
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    {
      "title": "NIST DLMF §7.2 — Error Function Definitions",
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      "url": "https://dlmf.nist.gov/7.2",
      "supports": "Definitions of erf and erfc and their complement relationship.",
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      "publisher": "National Institute of Standards and Technology"
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      "title": "NIST DLMF §5.2 — Gamma Function Definitions",
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      "url": "https://dlmf.nist.gov/5.2",
      "supports": "Definition of the gamma function and its poles.",
      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
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      "title": "NIST DLMF §5.12 — Beta Function",
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      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
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      "title": "NIST DLMF §8.17 — Incomplete Beta Functions",
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      "url": "https://dlmf.nist.gov/8.17",
      "supports": "Incomplete and regularized incomplete beta functions and their parameter domains.",
      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
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      "title": "NIST DLMF §10.2 — Bessel Function Definitions",
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      "url": "https://dlmf.nist.gov/10.2",
      "supports": "Bessel functions J and Y and their defining differential equation.",
      "kind": "reference",
      "publisher": "National Institute of Standards and Technology"
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      "title": "NumPy — numpy.sinc",
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      "supports": "Normalized sinc sin(πx)/(πx), the value 1 at zero, and conversion to the unnormalized sin(x)/x convention. This documents definitions, not a claim that CalculatorX calls NumPy. Mathematical reference only; not validation or endorsement of this implementation.",
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