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

instrument mechanism / verified-clean

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

The inspectable method record for System of Equations Solver (n×n linear), kept separate from implementation and qualification.

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

## Model scope

The mathematical method is valid for exact linear algebra. NEEDS_REVIEW is required because the specification leaves numerical tolerance, pivoting, supported domain, input validation, and infinite-solution parameterization ambiguous. The user-facing label should state whether results are exact or floating-point and should disclose behavior for singular and nearly singular systems.

- Gaussian elimination with back-substitution and rank comparison is the standard correct method for linear systems, but the specification does not define the rank/classification criteria explicitly: unique iff rank(A)=rank([A|b])=n, none iff rank(A)<rank([A|b]), and infinite iff rank(A)=rank([A|b])<n.

- Pivoting strategy and numerical rank tolerance are unspecified, so results for floating-point or nearly singular systems are ambiguous.

- The numeric domain is unspecified: real-only, complex-valued, exact rational, and floating-point inputs can produce different behavior and parameterization requirements.

- The parameter naming and free-variable ordering for infinite-solution outputs are unspecified.

## Units and conventions

- Matrix entries are assumed to be real, dimensionless numeric values unless the implementation explicitly supports complex or unit-bearing values.

- For dimensional systems, every term in each equation must have compatible units; coefficient units multiplied by variable units must match the corresponding right-hand-side units.

- No unit conversion is performed by Gaussian elimination; all supplied values must already use compatible units.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "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]."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

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.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/system-of-equations-solver-n-n-linear/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/system-of-equations-solver-n-n-linear/implementation/>)[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`

- `BOUNDARY_HAZARD_CATALOG.json`