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Field notes / Experiment guide

Is cube-root growth really shared by different ants?

Many chaotic ants appear to grow in width like the cube root of time. Do their apparent differences survive much longer observation?

New to these ants? Start here.

An ant moves on a grid of coloured squares. Each colour tells it to turn left (L) or right (R). It advances that square to the next colour, then steps forward. A rule such as RLR is the list of turns, starting at colour zero. On a blank grid the same rule always makes the same trajectory.

Territory means every cell visited so far. Its boundary consists of visited cells with an unvisited neighbour. A highway is a repeating pattern of motion that carries the ant away. A tether means repeated returns to a small home region.

Growing like the cube root of time means taking eight times as many steps to double a linear size. The rotor walker is a comparison model that follows and rotates arrows stored on cells. A seed makes a random starting pattern reproducible; the ant itself can still move deterministically.

We distinguish c, the chance of discovering a cell per boundary step, from kappa (κ), time at the boundary normalized by perimeter divided by area. Explaining their values is harder than using them to account for growth.

A closer look

Distributions of fitted width and gyration-radius exponents for compact ant rules and rotor controls, with a one-third reference line.
The crowded peak near one third is the puzzle. Each histogram summarizes finite-window fits; the rotor control shows why a fitted peak is evidence to investigate, rather than a proof of an exact exponent. Census ants are single runs per rule, while rotor samples vary the initial arrows. Figure source and recipe ↗
RLRLLL switches from expensive blob growth to column sweeping; the lower panels show a period-three wake, back-and-forth motion and two sweep predictions.
Read the upper panel as the cost of a new cell: after the late switch it drops sharply. Below, the colour wake and sawtooth motion reveal the column sweep. The two prediction checks describe this phase of one deterministic trajectory; they do not identify what triggers it. Figure source and recipe ↗

The idea under test

If the differences are transients, long-run ant exponents should approach the rotor control rather than settle at distinct values.

What we learned

Under the revised, well-posed comparison, all evaluable ants match the rotor at the late window. RLRLLL instead breaks out into a different growth structure.

What this leaves open

Matching a finite-window control does not establish an exact asymptotic exponent. Escapes and changes of regime are part of the result.

Read the evidence and reproduction recipe ↗

Inside the experiment

Methods, registered predictions, exact commands and saved analysis. These are the working research documents; expand them for the detail behind the introduction.

Methods and original experiment record

b1-exact-exponent

Migrated legacy investigation (2026-10-14). Its files used to sit in the flat data/ directory (and calib/, occ/, snaps/, dumps*, knock* where listed) with the label prefix below; experiments/_migration/map.tsv gives the exact old -> new path of every file. This stub was generated from file names and script docstrings. It is not the original write-up: the question, timestamped hypotheses, commands and results are in notes/ (mainly notes/frontier-results.md, notes/frontier-status.md, notes/research-frontier*.md; search for the label), and the producing command is usually in the script docstring or a run_*.sh here.

Old labels / prefixes: b1, b1b, run

Scripts

  • scripts/b1_select.py: B1/A2/A4 rule set: the 6 rules already run to 2^32 (experiments/2026-10-13-probe-growth-census/results/long.csv) + the 11 item-4 rules (13 distinct after overla
  • scripts/b1_stats.py: B1 and A2: 20 ants + 8 rotors run to 2^34 (experiments/2026-10-13-b1-exact-exponent/results/b1_probe_NN.csv). Design and decision rules: notes/frontier-2026-10-
  • scripts/b1b_stats.py: B1b: patched-start ants to 2^34 (seeds 301-303), criterion vs the rotor (mean 0.6676, seed sd 0.0063): |mean a_inf - 0.6676| < 0.0145. Design: notes/frontier-20
  • scripts/run_b1.sh: B1/A2: 20 ants + 8 rotors to 2^34. usage (WSL): RAYON_NUM_THREADS=3 bash experiments/2026-10-13-b1-exact-exponent/scripts/run_b1.sh
  • scripts/run_b1b.sh

Results

46 entries in results/ (22 are .log/.done markers); csv/json/txt outputs, *_stats.txt hold the summaries quoted in the notes.

Notebook

No committed Jupyter notebook is available for this experiment yet.

2026-10-13-b1-exact-exponent · Source revision ba38590b33e3

Original source ↗ · Input hashes

Introductions summarize the evidence; the expandable record preserves the original predictions, methods and caveats. Figures and notebooks are published as saved. Publication runs no simulations or notebook cells.