Coordinate Geometry Toolkit mechanism

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

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