Source: https://guthlabs.ai/docs/instruments/system-of-equations-solver-n-n-linear/implementation/

instrument implementation / implementation-contract

# System of Equations Solver (n×n linear) implementation

A source-bound implementation contract for System of Equations Solver (n×n linear); 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**: t8-018

**Effect**: Computation or source lookup only

## Input and validation requirements

The specification does not state the augmented-column sign convention, allowing two plausible interpretations with different solution vectors. Singular systems also expose a zero-pivot hazard: the solver must classify rank deficiency before back-substitution rather than divide by zero.

## 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": {
    "augmented_matrix": [
      [
        2,
        1,
        5
      ],
      [
        1,
        -1,
        1
      ]
    ]
  },
  "expected_outputs": {
    "classification": "unique",
    "solution": [
      2,
      1
    ],
    "rank_coefficient_matrix": 2,
    "rank_augmented_matrix": 2,
    "verification_residual": [
      0,
      0
    ]
  },
  "reference_data": {
    "system": [
      "2x + y = 5",
      "x - y = 1"
    ]
  },
  "working": "Start with 2x+y=5 and x-y=1. Subtract the second equation from the first: (2x+y)-(x-y)=5-1, so x+2y=4. Add this equation to the second equation: (x+2y)+(x-y)=4+1, so 2x+y=5, which is unchanged; instead solve directly from x-y=1, giving x=y+1. Substitute into 2x+y=5: 2(y+1)+y=5, so 3y=3 and y=1. Then x=2. The coefficient determinant is 2(-1)-1(1)=-3, nonzero, so rank(A)=2 and the solution is unique. Verification: [2(2)+1-5, 2-1-1]=[0,0]."
}
````

## Other records

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