Source: https://guthlabs.ai/docs/instruments/diophantine-equation-solver-linear/mechanism/

instrument mechanism / verified-with-binding-errata

# Diophantine Equation Solver (linear) mechanism

The inspectable method record for Diophantine Equation Solver (linear), kept separate from implementation and qualification.

Status: **VERIFIED_WITH_ERRATA**Lifecycle: **verified-with-binding-errata**

## Model scope

The stated divisibility criterion is mathematically correct for (a,b)!=(0,0). The standard complete construction is: find integers u,v with au+bv=d, set x0=u(c/d), y0=v(c/d), then x=x0+(b/d)t and y=y0-(a/d)t for t in Z. The specification needs to state this construction and define the (0,0) case. The fourth test vector's expected output intentionally records the corrected standard parameterization issue and therefore should be fixed before implementation.

- The existence criterion gcd(a,b)|c is correct for integers a and b not both zero, but the specification omits the explicit formulas for constructing a particular solution and the complete general solution.

- The specification does not define the degenerate case a=0 and b=0: if c=0, every integer pair is a solution; if c is nonzero, no solution exists.

- The sign convention and required positivity for gcd(a,b) are unspecified; the standard convention is d=gcd(a,b)>0.

## Units and conventions

- a, b, and c are exact integers with no physical units.

- The parameter t in the general solution is any integer.

- The standard gcd convention gcd(a,b)>=0 is assumed, with d>0 when (a,b) is not (0,0).

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "a": 15,
    "b": 21,
    "c": 6
  },
  "expected_outputs": {
    "gcd": 3,
    "exists": true,
    "particular": {
      "x0": -1,
      "y0": 1
    },
    "general_solution": "x=-1+7t, y=1-5t, t in Z"
  },
  "working": "gcd calculation: 21=1*15+6; 15=2*6+3; 6=2*3+0, so d=3. A particular solution is x0=-1, y0=1: 15*(-1)+21*(1)=-15+21=6. General solution: x=x0+(b/d)t=-1+(21/3)t=-1+7t; y=y0-(a/d)t=1-(15/3)t=1-5t, with t an integer."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The zero-coefficient boundary is solvable and must preserve a free variable, while the (0,0,1) input makes gcd(0,0) undefined and requires explicit rejection. A naive implementation that divides by gcd without guarding this case can fail or return invalid output.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/diophantine-equation-solver-linear/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/diophantine-equation-solver-linear/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/diophantine-equation-solver-linear/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`

- `BOUNDARY_HAZARD_CATALOG.json`