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    "Typing computes locally. Shared URLs send their parameters to the server for SSR and may appear in logs; REST/MCP also receive the inputs. Share links run the current engine, not a historical runtime.",
    "Quotient mode divides in IEEE-754 binary64, then rounds the display with Number.prototype.toFixed (round half away from 0 on that binary64 value). Trailing zeros may be stripped.",
    "The displayed decimal is not claimed to be an exact repeating expansion.",
    "REST field exact is true iff ieee_value equals the real number a÷b for those binary64 operands. It is not display round-trip and not integer divisibility. Example: (2^53−1)÷7 has remainder 3, so exact is false even when toFixed round-trips.",
    "REST field display_exact is true iff Number(ieee_value.toFixed(n)) === ieee_value (display did not change the binary64 value).",
    "Long division uses exact BigInt Euclidean division for non-negative integers: a, b ∈ ℤ≥0, b > 0. Integers are limited to 48 decimal digits to bound computation and long-division step output (a CalculatorX product contract, not a JavaScript BigInt limit).",
    "Remainder satisfies 0 ≤ r < b in long division (verified as b·q + r = a in BigInt). Long-mode exact is true iff remainder is 0.",
    "Quotient mode does not silently round normalized integral operands outside ±(2^53−1); those inputs return INTEGER_OUT_OF_SAFE_RANGE. Send large integers as decimal strings and use Long division for an exact result.",
    "REST/JSON reports ieee_value (raw binary64, or null when the long-division operands do not fit in a safe integer), decimal_value (display string), precision, rounding, exact (mathematical vs binary64 operands), display_exact (toFixed round-trip), exact_value (a/b when both are integers), integer_quotient/remainder for integer quotient-mode operands, and integer_engine=bigint in long mode."
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    "All integral quotient operands must fit±(2^53−1). Use Long mode with non-negative integer strings for larger values, up to48 digits.",
    "Tokens at most512 characters; unknown fields, mode/type errors and conflicting aliases rejected. Nonzero text/JSON/result underflow is rejected; subnormal nonzero values accepted.",
    "Division by zero and overflow rejected. Decimal places default8 when absent/blank, otherwise integer0–16.",
    "Share URLs restore current-engine inputs; not a historical runtime."
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  "boundary_conditions": [
    "Accept finite decimal numbers or decimal/scientific strings up to512 characters. Unknown fields, conflicting aliases and unsupported modes are rejected. Quotient mode rejects all normalized integral operands outside the safe-integer range; use non-negative integer strings in Long mode. JSON numbers already reflect client binary64 rounding.",
    "b = 0 → DIVISION_BY_ZERO",
    "Non-finite a or b → INVALID_NUMBER / MISSING_REQUIRED_INPUT",
    "Non-finite binary64 quotient (overflow) → INVALID_NUMBER; nonzero input or quotient rounded to zero → NUMERIC_UNDERFLOW",
    "Long mode: negatives or non-integers → INVALID_INPUT (use Quotient mode)",
    "Long mode: more than 48 digits → INTEGER_TOO_LONG (product limit to bound computation and step output)",
    "Quotient mode: normalized integral operand outside ±(2^53−1) → INTEGER_OUT_OF_SAFE_RANGE",
    "decimals missing or blank → default 8; null and non-decimal tokens rejected; otherwise must be an integer in 0…16 or DECIMAL_PLACES_OUT_OF_RANGE (no silent clamp)"
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  "sources": [
    {
      "title": "ECMAScript Language Specification",
      "section": "Number.prototype.toFixed ( fractionDigits )",
      "url": "https://tc39.es/ecma262/#sec-number.prototype.tofixed",
      "supports": "Quotient-mode display rounding after binary64 division",
      "kind": "standard"
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      "title": "ECMAScript Language Specification",
      "section": "Number type (IEEE-754 binary64)",
      "url": "https://tc39.es/ecma262/#sec-ecmascript-language-types-number-type",
      "supports": "Quotient mode uses the ECMAScript Number type, which is IEEE-754 binary64",
      "kind": "standard"
    },
    {
      "title": "IEEE Std 754-2019",
      "section": "IEEE Standard for Floating-Point Arithmetic",
      "url": "https://standards.ieee.org/ieee/754/6210/",
      "supports": "Binary floating-point format used by quotient mode",
      "kind": "standard"
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    {
      "title": "MIT Mathematics for Computer Science — Chapter 8: Number Theory",
      "section": "Chapter 8, including sections 8.2–8.9 and problem 8.24",
      "url": "https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-spring-2015/mit6_042js15_textbook.pdf",
      "supports": "Divisibility, prime factorization, gcd and the Euclidean algorithm, lcm via prime exponents, division with remainder, modular congruence and multiplicative inverses. CalculatorX’s signed-remainder and zero-input policies are stated separately on the page. Mathematical reference only; not validation or endorsement of this implementation.",
      "kind": "reference",
      "publisher": "Massachusetts Institute of Technology"
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    {
      "title": "Margaret Stevens — The Maths Workbook (University of Oxford)",
      "section": "Chapter 1, §1: Algebraic Expressions and Fractions; §3: Equations",
      "url": "https://www.some.ox.ac.uk/wp-content/uploads/2024/08/Maths-Workbook.pdf",
      "supports": "Algebraic addition/subtraction, multiplication/division, distributive rules, equivalent fractions, common denominators and reciprocals. Ratios are interpreted as quotients. Equality of ratios with nonzero denominators permits cross multiplication, and solving a proportion uses the chapter’s equation transformations. A zero divisor is excluded; integer quotient/remainder uses the separately cited Euclidean-division reference. Mathematical reference only; not implementation validation or an endorsement of CalculatorX.",
      "kind": "reference",
      "publisher": "University of Oxford, Department of Economics; hosted by Somerville College"
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