Source: https://guthlabs.ai/docs/instruments/coordinate-geometry-toolkit/mechanism/

instrument mechanism / verified-clean

# Coordinate Geometry Toolkit mechanism

The inspectable method record for Coordinate Geometry Toolkit, kept separate from implementation and qualification.

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

## Model scope

The explicitly supplied distance and point-line-distance formulas are correct under the stated Euclidean and unit-consistency assumptions. The instrument cannot receive a PASS until formulas and constraints for slope, line equation, and circle-from-three-points are specified, and the empty honest_label is corrected to disclose limitations and degeneracy handling.

- The distance formula is standard and correct for two points in a common Euclidean coordinate system.

- The midpoint formula is only stated as 'midpoint avg'; it must explicitly mean coordinate-wise averaging: ((x1+x2)/2, (y1+y2)/2).

- The point-line distance formula is standard and correct, but requires a^2+b^2>0.

- The instrument claims slope computation, but no slope formula is provided; the standard formula is (y2-y1)/(x2-x1), with x2!=x1 required.

- The instrument claims a line equation from two points, but no formula or vertical-line handling is provided; the two points must be distinct.

- The instrument claims a circle from three points, but no formula is provided; the three points must be non-collinear for a unique circle.

## Units and conventions

- All point coordinates in a single distance computation must use the same length unit.

- For midpoint and line-equation calculations, corresponding coordinates must use compatible units.

- In ax+by+c, the coefficients must be dimensionally compatible with the coordinate units; otherwise the resulting distance unit is ambiguous.

- The coordinate system is assumed to be ordinary Euclidean Cartesian coordinates.

- No unit conversion is performed automatically.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "operation": "distance_and_midpoint",
    "point1": {
      "x": 1,
      "y": 2
    },
    "point2": {
      "x": 4,
      "y": 6
    },
    "unit": "m"
  },
  "expected_outputs": {
    "distance": 5,
    "midpoint": {
      "x": 2.5,
      "y": 4
    },
    "distance_unit": "m"
  },
  "working": "dx = x2 - x1 = 4 - 1 = 3. dy = y2 - y1 = 6 - 2 = 4. distance = sqrt(dx^2 + dy^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5 m. midpoint = ((x1+x2)/2, (y1+y2)/2) = ((1+4)/2, (2+6)/2) = (5/2, 8/2) = (2.5, 4)."
}
````

## Failure boundary

- Slope division by zero when x2=x1; vertical lines need a separate representation.

- Line equation is undefined or non-unique when the two input points coincide, and vertical lines cannot be represented safely as y=mx+b.

- Point-line distance divides by zero when a=0 and b=0; this input must be rejected.

- A unique circle does not exist for three collinear points, and coincident or insufficiently distinct points create degeneracies.

- Mixed coordinate units, such as one point in meters and another in centimeters, produce incorrect distances unless converted first.

- Finite-precision arithmetic can cause near-collinear three-point circle inputs to be unstable.

- The spec does not define output formats for vertical lines, undefined slopes, degenerate circles, or invalid inputs.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/coordinate-geometry-toolkit/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/coordinate-geometry-toolkit/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/coordinate-geometry-toolkit/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-tranche4/tranche4-results.json`