Source: https://guthlabs.ai/docs/instruments/polar-curve-plotter/

instrument / verified-with-binding-errata

# Polar Curve Plotter

The current ledger marks this instrument verified with errata that must be applied at binding time.

Status: **VERIFIED_WITH_ERRATA**Lifecycle: **verified-with-binding-errata**

## Inspectable records

[01**Overview**Purpose and lifecycle](<https://guthlabs.ai/docs/instruments/polar-curve-plotter/>)[02**Mechanism**Method and assumptions](<https://guthlabs.ai/docs/instruments/polar-curve-plotter/mechanism/>)[03**Implementation**Binding and validation](<https://guthlabs.ai/docs/instruments/polar-curve-plotter/implementation/>)[04**Qualification**Evidence and errata](<https://guthlabs.ai/docs/instruments/polar-curve-plotter/qualification/>)

## Ledger state

The current ledger marks this instrument verified with errata that must be applied at binding time.

**Area**: Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)

**Tranche**: tranche-8

**Evidence ID**: t8-027

**Ledger rows**: 1

No ledger note is attached.

## Review findings

The stated polar area integrand is mathematically correct under radians and a properly selected, non-overcounting tracing interval. The instrument needs an explicit angle-unit contract and a distinction between swept area and geometric enclosed area before receiving a PASS verdict.

### Formula and scope findings

- The polar area formula A=∫(1/2)r(θ)^2 dθ is standard for swept area over a correctly chosen interval, but it does not always equal the geometric area enclosed by a self-intersecting or multiply traced curve; the specification must define whether multiplicity is counted or overlapping regions are unioned.

- Sampling over θ is insufficiently specified: the implementation needs an adaptive or resolution-controlled sampling strategy to avoid aliasing narrow loops, cusps, rapidly oscillating curves, and discontinuities.

- The specification does not state whether θ is measured in radians or degrees. The calculus formula requires θ in radians unless an explicit degree-to-radian conversion is applied.

- The specification does not define how the enclosed-area interval is selected when the supplied θ range contains only part of a curve, multiple revolutions, or retraced portions.

### Unit assumptions

- All θ values in the formulas and test vectors are radians unless the input explicitly declares degrees.

- r has the same distance unit as the requested plotted coordinate system; area is therefore in squared distance units.

- For an explicitly degree-based range, the implementation must convert dθ using dθ_radians=(π/180)dθ_degrees before applying the polar area formula.

- The area results below assume the supplied interval traces the intended curve exactly once, except where the interval is explicitly identified as a partial trace.

## Boundary hardening

**BREAK_CLAIMED** · SELF_FLAGGED

The singular expression r(theta)=1/(theta-pi/2) produces an undefined plot value and divergent area when the requested range includes theta=pi/2. The degree-versus-radian convention probe also produces materially different numeric output unless units are enforced explicitly.

## Errata and corrections

### Tranche errata

**enclosed_area_exact / numeric** · vector 1

2π ≈ 6.2832

4π ≈ 12.5664

area of a circle of radius 2 (r(θ)=2 constant) is π·r²=4π; her own working evaluates 1/2·4·(2π) and writes the intermediate product as "4π/2" instead of 4π, dropping a factor of 2 in the polar-area integral ∫½r²dθ

## Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

````
{
  "inputs": {
    "r_theta": "2",
    "theta_range": [
      0,
      "2π"
    ],
    "theta_units": "radians",
    "area_units": "square distance units"
  },
  "expected_outputs": {
    "plot": "Circle centered at the pole with radius 2",
    "enclosed_area_exact": "2π",
    "enclosed_area_numeric": 6.283185307179586,
    "numeric_quadrature_area": 6.283185307179586
  },
  "reference_data": {
    "identity": "A=1/2∫_0^(2π)4 dθ=2π"
  },
  "working": "A=1/2∫_0^(2π)(2)^2 dθ. First, (2)^2=4. Then A=1/2·4·(2π)=4π/2=2π=6.283185307179586."
}
````

## Related workflows

No source-defined multi-instrument workflow membership is recorded.

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json · VERIFIED_WITH_ERRATA`

- `evidence-tranche8/tranche8-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`