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

instrument mechanism / verified-clean

# Linear Equation Solver mechanism

The inspectable method record for Linear Equation Solver, kept separate from implementation and qualification.

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

## Model scope

The stated algorithms are mathematically standard, but the instrument needs explicit domain, dimensional-unit, parser, singular-system, and numerical-tolerance policies before receiving an unconditional PASS.

- Substitution and elimination are standard correct methods for linear equations and systems.

- Cramer's rule is correct only for square systems with a nonzero determinant; the specification does not state how singular, dependent, or inconsistent systems are handled.

- The supported variable count, equation count, and coefficient-shape constraints are not explicitly defined for the claimed 2-3 equation systems.

- Free-text equation parsing is underspecified, including variable-name rules, implicit multiplication, signs, parentheses, malformed input, and nonlinear terms.

## Units and conventions

- All terms added or equated within an equation must have compatible physical units.

- For ax+b=c, a, b, and c are assumed dimensionally compatible with x and the equation's result; if x has units, a generally carries result-units per x-unit.

- No unit conversion is performed unless explicitly implemented; coefficients and constants are otherwise treated as dimensionless numbers.

- All coefficients and constants are assumed to be finite real numbers unless complex or symbolic domains are explicitly supported.

## Inspectable vector

One of 4 source-bound review vectors:

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

## Failure boundary

- 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.

## Other records

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