code/distributed_scanning/tangled_loops/creation.md

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

{
  "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

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