Source: https://guthlabs.ai/docs/instruments/slope-field-direction-field-plotter/mechanism/

instrument mechanism / verified-clean

# Slope Field / Direction Field Plotter mechanism

The inspectable method record for Slope Field / Direction Field Plotter, kept separate from implementation and qualification.

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

## Model scope

The stated slope-field method is mathematically standard, and RK4 is the standard fourth-order explicit Runge-Kutta method, but the specification is incomplete for deterministic verification. Exact ambiguities requiring resolution are: plot x/y bounds, grid-point placement, segment length, segment normalization, RK4 step size, integration direction, trajectory stopping/clipping rules, and non-finite-expression handling.

- The core direction-field formula, slope=f(x,y), is standard and correct.

- The RK4 claim is standard but underspecified: the four stage equations, step size, integration direction, domain limits, and handling of failed evaluations are not defined.

- Grid density alone does not define a grid; x/y bounds and grid-point placement are missing.

- Line-segment normalization and segment length are unspecified. A slope field may use (1,f), a normalized vector, or another scaling convention, producing different plots.

- Initial-condition overlays require explicit initial x/y coordinates, integration direction, step size, stopping criteria, and whether trajectories are clipped to the plot domain.

- The implementation must reject or handle non-finite f(x,y) values, discontinuities, poles, and expressions outside their domains.

## Units and conventions

- x and y are treated as numerical coordinates; no physical units are specified.

- For dy/dx=f(x,y), f has units of y-units per x-unit when x and y carry physical units.

- The test vectors use dimensionless x and y, so slopes are dimensionless.

- If x and y use different physical units, the plotting implementation must preserve the resulting slope units; no unit-conversion behavior is specified.

- Reference calculations below use centered field segments of total geometric length 0.2 and the normalized tangent (1,m)/sqrt(1+m^2). This convention is an explicit test assumption, not defined by the instrument spec.

## Inspectable vector

One of 4 source-bound review vectors:

````
{
  "inputs": {
    "f": "x+y",
    "grid_point": {
      "x": 1,
      "y": 2
    },
    "segment_total_length": 0.2,
    "render_mode": "direction_field_only"
  },
  "expected_outputs": {
    "slope": 3,
    "normalized_tangent": {
      "dx": 0.31622776601683794,
      "dy": 0.9486832980505138
    },
    "segment_endpoints": [
      {
        "x": 0.9683772233983162,
        "y": 1.9051316701949486
      },
      {
        "x": 1.0316227766016839,
        "y": 2.094868329805051
      }
    ]
  },
  "reference_data": {
    "segment_rule": "centered endpoints p - 0.1*(1,m)/sqrt(1+m^2) and p + 0.1*(1,m)/sqrt(1+m^2)"
  },
  "working": "At (x,y)=(1,2), m=f(1,2)=1+2=3. Compute sqrt(1+m^2)=sqrt(10)=3.1622776601683793. The unit tangent is (1/sqrt(10),3/sqrt(10))=(0.31622776601683794,0.9486832980505138). Half the segment length is 0.2/2=0.1. The offset is (0.0316227766016838,0.0948683298050514). Subtracting from (1,2) gives (0.9683772233983162,1.9051316701949486); adding gives (1.0316227766016838,2.0948683298050514)."
}
````

## Failure boundary

**BREAK_CLAIMED** · SELF_FLAGGED

The input f(x,y)=1/(x-y) with an initial or grid point at (0,0) produces division by zero. The plotter must detect the singularity and reject or omit that point rather than emitting a finite slope or attempting an RK4 step.

## Other records

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