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

instrument implementation / implementation-contract

# Inequality Solver & Number-Line Grapher implementation

A source-bound implementation contract for Inequality Solver & Number-Line Grapher; 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**: t6-016

**Effect**: Computation or source lookup only

## Input and validation requirements

- 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.

## 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": {
    "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, ∞)."
}
````

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/inequality-solver-number-line-grapher/>)[02**Mechanism**](<https://guthlabs.ai/docs/instruments/inequality-solver-number-line-grapher/mechanism/>)[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`