Source: https://guthlabs.ai/docs/instruments/conic-section-identifier-plotter/mechanism/

instrument mechanism / verified-clean

# Conic Section Identifier & Plotter mechanism

The inspectable method record for Conic Section Identifier & Plotter, kept separate from implementation and qualification.

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

## Model scope

NEEDS_REVIEW because the specification does not define coefficient/coordinate units, degenerate-conic behavior, numerical tolerances, plot conventions, or the complete rotated-conic algorithm. The supplied discriminant and completion-of-square description is insufficient for the claimed general equation, especially when B≠0. Test vector 3 intentionally reports the corrected hand calculation in expected_outputs and flags the erroneous 5-value in working; the vector should be repaired before execution.

- B²-4AC classifies nondegenerate real conics as ellipse (<0), parabola (=0), or hyperbola (>0), but it does not by itself identify circles, degenerate conics, empty real loci, point conics, or pairs of lines.

- For B≠0, completion of squares requires a coordinate-axis rotation and removal of the xy term; the stated formula set does not provide that rotation, its angle, or the transformed coefficients.

- The conic type must be checked against the full quadratic and augmented matrices to detect degeneracy before reporting foci, vertices, eccentricity, directrices, or a plot.

- Foci, vertices, eccentricity, and directrices require separate formulas for ellipse, hyperbola, and parabola; those formulas are not specified.

- For a circle, eccentricity is 0 and the usual focus/directrix definition is degenerate; the output convention must explicitly define whether the coincident center is reported as a limiting focus and whether directrix is reported as undefined.

## Units and conventions

- x and y are assumed to have the same coordinate unit.

- A, B, C, D, E, and F must have compatible units so every term in Ax²+Bxy+Cy²+Dx+Ey+F has the same unit; if x and y are dimensionless, all coefficients may be treated as dimensionless.

- Reported lengths such as radius, semimajor axis, focal distance, and directrix distance use the coordinate unit.

- Plotting requires a declared axis scale and aspect ratio; unequal visual scaling can distort the apparent conic.

- The sign of B²-4AC is invariant under multiplying the entire equation by any nonzero scalar, but numerical tolerance for values near zero must be specified.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "A": 1,
    "B": 0,
    "C": 1,
    "D": -4,
    "E": 6,
    "F": -3
  },
  "expected_outputs": {
    "conic_type": "circle",
    "standard_form": "(x-2)^2+(y+3)^2=16",
    "center": [
      "2",
      "-3"
    ],
    "radius": "4",
    "eccentricity": "0",
    "foci": [
      [
        "2",
        "-3"
      ]
    ],
    "vertices": [
      [
        "6",
        "-3"
      ],
      [
        "-2",
        "-3"
      ],
      [
        "2",
        "1"
      ],
      [
        "2",
        "-7"
      ]
    ],
    "directrices": "undefined for a circle under the standard focus/directrix definition",
    "plot_locus": "circle centered at (2,-3) with radius 4"
  },
  "reference_data": {
    "classification_discriminant": "B²-4AC",
    "standard_circle_form": "(x-h)²+(y-k)²=r²"
  },
  "working": "Discriminant: B²-4AC = 0²-4(1)(1) = -4 < 0, so the nondegenerate conic is ellipse-type; equal positive quadratic coefficients identify a circle. Complete squares: x²-4x=(x-2)²-4 and y²+6y=(y+3)²-9. Therefore (x-2)²-4+(y+3)²-9-3=0, so (x-2)²+(y+3)²=16. Center=(2,-3), radius=sqrt(16)=4, eccentricity=0. The four axis vertices are (2±4,-3) and (2,-3±4)."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The all-zero coefficient input produces discriminant zero but is not a conic and causes zero-denominator or undefined-parameter calculations. The instrument must reject it instead of classifying it as a parabola. Rotated-conic outputs also require an explicit rotation convention because equivalent angle choices can swap axis labels and signs.

## Other records

[01**Overview**](<https://guthlabs.ai/docs/instruments/conic-section-identifier-plotter/>)[03**Implementation**](<https://guthlabs.ai/docs/instruments/conic-section-identifier-plotter/implementation/>)[04**Qualification**](<https://guthlabs.ai/docs/instruments/conic-section-identifier-plotter/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`

- `BOUNDARY_HAZARD_CATALOG.json`