Source: https://guthlabs.ai/docs/instruments/inequality-solver-number-line-grapher/mechanism/

instrument mechanism / verified-clean

# Inequality Solver & Number-Line Grapher mechanism

The inspectable method record for Inequality Solver & Number-Line Grapher, kept separate from implementation and qualification.

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

## Model scope

The underlying sign-chart method is valid for real linear, polynomial, and rational inequalities. The instrument remains NEEDS_REVIEW because its parser grammar, supported expression subset, real-domain policy, numerical tolerance, and rendering semantics are unspecified. The four vectors use exact real arithmetic and include a rational denominator-zero boundary at x=-2.

- Sign-chart/interval testing is a standard method, but the specification does not define how expressions are parsed or normalized before sign analysis.

- The solver must identify every zero of the numerator or polynomial and every undefined point from a denominator, preserve multiplicities, and apply strict versus non-strict inequality endpoints correctly.

- Rational expressions require domain exclusions even when an algebraic factor could be canceled; canceled denominator zeros remain holes in the domain.

- The specification does not state whether coefficients and variables are restricted to real numbers, which is required for ordinary number-line inequality solutions.

- The honest_label field is empty; the instrument should state that it solves real-valued inequalities and that interval endpoints, holes, and infinities are represented using standard interval notation.

## Units and conventions

- All variables and constants are real-valued and dimensionally compatible.

- If units are supplied, every additive term must have the same units; coefficients multiplying x must have units that make the term compatible with the constant term.

- The number line is dimensionless unless the user explicitly supplies a common unit.

- Infinity is a mathematical interval endpoint, not a finite numeric value.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "inequality": "2x - 6 > 0",
    "sampled_x_values": [
      2,
      3,
      4
    ]
  },
  "expected_outputs": {
    "interval_notation": "(3, ∞)",
    "number_line": "open circle at x=3; shade to the right",
    "sampled_points": [
      {
        "x": 2,
        "expression_value": -2,
        "satisfies": false
      },
      {
        "x": 3,
        "expression_value": 0,
        "satisfies": false
      },
      {
        "x": 4,
        "expression_value": 2,
        "satisfies": true
      }
    ],
    "curve_features": {
      "critical_points": [
        3
      ],
      "roots": [
        3
      ]
    }
  },
  "working": "Set the expression equal to zero: 2x - 6 = 0, so 2x = 6 and x = 3. Test x=2: 2(2)-6 = 4-6 = -2, which is not greater than 0. Test x=4: 2(4)-6 = 8-6 = 2, which is greater than 0. At x=3 the value is 0, and strict > excludes it. Therefore the solution is (3, ∞)."
}
````

## Failure boundary

- Division by zero must be detected before evaluation; denominator zeros are excluded even if they are not roots of the numerator.

- Strict inequalities require open endpoints, while <= and >= include valid finite zeros; a denominator zero is never included.

- Repeated roots may not change the sign across a critical point, so the implementation must use multiplicity or test each interval rather than blindly alternating signs.

- Floating-point arithmetic can misclassify values near roots or denominator zeros; exact rational arithmetic or tolerance-aware endpoint handling is preferable.

- Expressions with malformed syntax, implicit multiplication, unsupported functions, or mixed real and complex values need explicit rejection or documented behavior.

- Unbounded intervals and disconnected unions must render correctly on the number line.

- A plot of a rational expression must distinguish a root, an excluded hole, and a vertical asymptote.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/inequality-solver-number-line-grapher/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/inequality-solver-number-line-grapher/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/inequality-solver-number-line-grapher/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-tranche6/tranche6-results.json`