Source: https://guthlabs.ai/docs/instruments/gcd-lcm-calculator/mechanism/

instrument mechanism / verified-clean

# GCD/LCM Calculator mechanism

The inspectable method record for GCD/LCM Calculator, kept separate from implementation and qualification.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Model scope

Core Euclidean and extended-Euclidean methods are mathematically correct, but the specification is not complete enough for an unambiguous production instrument. The honest label should state accepted cardinality, integer-only validation, sign conventions, zero conventions, and how coefficients are returned for 2+ inputs.

- The stated Euclidean algorithm correctly supports GCD computation, and the extended Euclidean algorithm correctly supports Bezout coefficients.

- The LCM formula is not explicitly specified. For two integers, use lcm(a,b)=|ab|/gcd(a,b), with a zero-result convention; for multiple integers, apply lcm iteratively or define an equivalent general rule.

- For 2+ integers, the meaning and ordering of multiple Bezout coefficients is not explicitly defined; they should satisfy a1*c1+...+an*cn=GCD(a1,...,an).

## Units and conventions

- All inputs are mathematical integers with no physical units.

- GCD is assumed to be the nonnegative greatest common divisor, including gcd(0,n)=|n|.

- LCM is assumed to be nonnegative, including lcm(0,n)=0 and lcm(0,0)=0 if all inputs are zero.

- For multiple inputs, Bezout coefficients correspond positionally to the input list and satisfy the linear combination equal to the positive GCD.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "integers": [
      84,
      30
    ]
  },
  "expected_outputs": {
    "gcd": 6,
    "lcm": 420,
    "bezout_coefficients": [
      -1,
      3
    ]
  },
  "working": "84 = 2*30 + 24; 30 = 1*24 + 6; therefore gcd = 6. Back-substitute: 6 = 30 - 24 = 30 - (84 - 2*30) = -1*84 + 3*30, so coefficients are [-1,3]. lcm = |84*30|/6 = 2520/6 = 420."
}
````

## Failure boundary

- Division by zero must be prevented when implementing lcm(a,b)=|ab|/gcd(a,b); the all-zero case requires an explicit convention.

- An empty input list is invalid and must be rejected because GCD, LCM, and Bezout coefficients are undefined without an agreed empty-list convention.

- A single integer conflicts with the stated 2+ input requirement and should be rejected or explicitly supported.

- Inputs must be validated as integers; fractional, NaN, infinity, and nonnumeric values are invalid.

- For multiple integers, naive multiplication can overflow fixed-width numeric types; arbitrary-precision integers or overflow-safe arithmetic may be required.

- Negative-input behavior must be specified, especially the sign of GCD and LCM and the nonuniqueness of Bezout coefficients.

- The all-zero input list needs an explicit policy: gcd(0,...,0) is commonly defined as 0, while Bezout coefficients are nonunique and the LCM convention varies.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/gcd-lcm-calculator/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/gcd-lcm-calculator/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/gcd-lcm-calculator/qualification/>)

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json`

- `evidence-tranche4/tranche4-results.json`