# Tangled Closed-Loop Counting

## Goal

Render a tangle of smooth **closed loops** on a single canvas. Every
loop is drawn in the **same** dark colour, so the model cannot recover
the count by colour-segmenting the image. Loops cross each other and
themselves freely, but never run parallel for long stretches — every
loop remains individually traceable.

This is a *distributed scanning* task: there is no single starting
point. The model must look across the whole image, follow each loop
all the way around, and report how many distinct closed loops exist.

## Differs from `sequential_traversal/tube_connection`

`tube_connection` labels endpoints and asks "which top label connects
to which bottom label", which is solved by tracing one path at a time.
Here there are no labels, no endpoints at all, and no per-loop tracing
question. The model needs the global *count* of distinct closed loops.

## Differs from `sequential_traversal/line_intersections`

`line_intersections` uses perimeter-anchored open curves and asks for
crossing sequences along one labelled string. Here every curve is a
closed loop (no endpoints anywhere), every loop is anonymous, all in
the same colour, and the only ground truth is the number of loops.

## Question

> How many distinct closed loops are tangled together in this image?
> Each loop is a single continuous curve that closes back on itself —
> there are no loose endpoints anywhere. All loops are drawn in the
> same colour and may cross other loops or themselves freely. Count
> the total number of distinct closed loops and report the count as a
> positive integer.

## Generation Procedure

1. **Sample N** in `[min_loops, max_loops]` (default 5–11).
2. **For each loop**, build a smooth closed curve:
   - Pick a random interior centre so the loop fits inside
     `interior_margin` (default 55 px) of the canvas edge.
   - Sample 6–10 waypoints on a jittered ring around that centre
     (angle jitter ≈ 0.42 rad, radius jitter ≈ 32 % of the mean
     radius, mean radius sampled from 150–275 px).
   - Fit a periodic cubic B-spline through the waypoints
     (`scipy.interpolate.splprep(..., per=True, k=3)`) and
     dense-sample 900 points along it. The last point is snapped to
     equal the first so the renderer draws a genuine closed curve.
3. **Find all crossings** (self and pair) — used only by the
   close-approach validator below.
4. **Reject** the sample if any two curves come within ~9 px of each
   other anywhere except at a real crossing point. For self-closeness
   the segment-index gap is computed **circularly** (with wraparound),
   since closed curves have no linear endpoint ordering.
5. **Render** all loops in the same dark colour onto an off-white
   canvas with subtle background noise.

## Anti-Shortcut Notes

The task has been through two shortcut-plugging rewrites:

1. **Unique-colour shortcut → single stroke colour.** An earlier
   version assigned each string a unique colour, collapsing the task
   to "count distinct colours". Fixed by drawing every curve in the
   same dark colour.
2. **Skeleton-endpoint shortcut → closed loops.** The previous
   perimeter-anchored design had each string as an *open* curve with
   two endpoints on the border. A ten-line attack recovered the
   answer on 28 / 30 samples: `skeletonize(gray < 110)` →
   count pixels with exactly one 8-neighbour → divide by two.
   Crossings are X/T junctions (degree ≥ 3), so only the two terminal
   endpoints of each string have degree 1, and endpoints ÷ 2 is the
   string count. Moving to closed curves eliminates the attack
   entirely — a closed loop's skeleton has no degree-1 pixels, so the
   attack returns 0 regardless of loop count. Verified on the
   regenerated dataset.
3. **No-near-parallel validation** (retained) prevents two loops that
   run parallel for a long stretch from reading as one fat loop,
   which would also confuse a human counter.

## Annotation Format

```json
{
  "image": "images/tangled_loops_00000.png",
  "width": 1024,
  "height": 1024,
  "num_loops": 7,
  "question": "How many distinct closed loops are tangled together ...",
  "answer": 7
}
```

## Output Organization

```text
tangled_loops/
  creation.py
  creation.md
  annotations.jsonl
  data.json
  images/
    tangled_loops_00000.png
    ...
```