Mandelbrot / Julia Set Explorer mechanism

Model scope

The recurrence is mathematically standard, but the specification is incomplete for a reproducible Mandelbrot/Julia renderer. Define mode-specific initialization, strict escape rule, iteration indexing, finite-depth semantics, coordinate-to-pixel mapping, and the escape-time coloring algorithm before claiming exact behavior.

• The recurrence zₙ₊₁=zₙ²+c is standard, but Mandelbrot and Julia initialization are unspecified: Mandelbrot requires z₀=0 with c equal to the sampled complex-plane point, while Julia requires z₀ equal to the sampled point with a fixed c parameter.
• The escape criterion is unspecified. The standard criterion is |zₙ|>2 after each iteration; using |zₙ|≥2 changes boundary classifications.
• Iteration numbering and whether the initial value z₀ is tested are unspecified.
• Escape-time coloring is not fully defined: the palette, color for non-escaping points, and whether smooth coloring or only integer escape time is used are missing.
• A finite iteration depth gives an approximation rather than an exact membership test: points not escaping by the depth must be labeled bounded-through-depth, not mathematically proven to lie in the set.
• The zoom region, pixel sampling convention, aspect-ratio handling, and complex-coordinate mapping are unspecified, so rendered pixels are not fully reproducible.

Units and conventions

• Complex coordinates and the parameter c are dimensionless real-imaginary pairs; no physical unit conversion applies.
• The escape radius is 2 in the complex modulus, equivalently 4 in squared modulus, and the implementation should compare |z|>2 or |z|²>4.
• Iteration depth is a nonnegative integer count of recurrence updates per sampled point.
• Zoom-region coordinates are assumed to be complex-plane coordinates, not pixel coordinates; pixel width and height are integer counts.
• The expected vectors assume Mandelbrot mode uses z₀=0 and Julia mode uses z₀ equal to the sampled point, with escape criterion |z|>2 checked after every update.

Inspectable vector

One of 4 source-bound review vectors:

{
  "inputs": {
    "mode": "Mandelbrot",
    "sample_point": {
      "x": 0,
      "y": 0
    },
    "c": {
      "x": 0,
      "y": 0
    },
    "z0": {
      "x": 0,
      "y": 0
    },
    "iteration_depth": 10,
    "escape_rule": "escape when |z|>2"
  },
  "expected_outputs": {
    "sampled_point": {
      "x": 0,
      "y": 0
    },
    "escape_status": "bounded_through_depth",
    "escape_iteration": null,
    "escape_time_color_input": "non-escaping"
  },
  "working": "Start z₀=0+0i and c=0+0i. z₁=z₀²+c=0²+0=0. z₂=z₁²+c=0²+0=0. Continuing gives zₙ=0 for n=0 through 10. |zₙ|=0≤2 at every check, so there is no escape iteration within depth 10."
}

Failure boundary

BREAK_CLAIMED · SELF_FLAGGED

Extreme finite inputs overflow JavaScript double arithmetic on the first iteration, producing Infinity. If the renderer does not reject or explicitly handle non-finite complex values, escape coloring and subsequent iterations can become invalid.

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-tranche6/tranche6-results.json
• BOUNDARY_HAZARD_CATALOG.json