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

instrument implementation / implementation-contract

# Linear Equation Solver implementation

A source-bound implementation contract for Linear Equation Solver; deployment and executable qualification remain separate.

Status: **VERIFIED_CLEAN**Lifecycle: **implementation-contract**

## Binding contract

This record describes what an executable surface must bind and validate. It does not claim that a public calculator, package, or production route is deployed.

**Record owner**: Guth Instruments

**Lifecycle**: verified-clean

**Evidence ID**: t8-008

**Effect**: Computation or source lookup only

## Input and validation requirements

- Division by zero occurs in the scalar solver when a=0; the cases 0*x+b=c must be classified as either no solution or infinitely many solutions.

- Cramer's rule must not be used when the determinant is zero; singular systems require rank-based classification into no solution or infinitely many solutions.

- A non-square coefficient matrix, underdetermined system, or overdetermined system needs an explicit solution-set policy.

- Floating-point arithmetic can misclassify nearly singular systems; determinant and residual tolerances must be specified.

- Free-text input can contain nonlinear expressions, contradictory variable names, omitted coefficients, or ambiguous syntax.

- The empty honest_label field should be populated to state supported domains, exact-versus-numeric behavior, singular-system behavior, and parser limitations.

## Required corrections

No binding erratum is attached to this unique-name record. Implementers must still preserve the declared units, domain, and failure behavior.

## Inspectable interface fixture

The sample below is evidence for interface and validation tests, not a production implementation.

````
{
  "inputs": {
    "equation": {
      "a": 3,
      "b": -5,
      "c": 10
    },
    "form": "a*x+b=c"
  },
  "expected_outputs": {
    "solution_set": {
      "x": "5"
    },
    "residual": "3*5-5-10=0",
    "spot_checks": [
      {
        "at": {
          "a": 3,
          "b": -5,
          "c": 10
        },
        "input_side": 5,
        "answer_side": 5
      },
      {
        "at": {
          "a": 4,
          "b": 7,
          "c": 19
        },
        "input_side": 3,
        "answer_side": 3
      }
    ]
  },
  "reference_data": {
    "method": "isolate x",
    "formula": "x=(c-b)/a",
    "domain_constraint": "a != 0"
  },
  "working": "3x-5=10; add 5 to both sides: 3x=15; divide by 3: x=15/3=5; residual: 3(5)-5-10=15-5-10=0. Numeric formula checks: (10-(-5))/3=15/3=5.00000000 and (19-7)/4=12/4=3.00000000."
}
````

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/linear-equation-solver/>)[02**Mechanism**](<https://guthlabs.ai/docs/instruments/linear-equation-solver/mechanism/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/linear-equation-solver/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-tranche8/tranche8-results.json`