rulepat: why do some rules paint a "plateau + diagonal rectangles" picture?

Trigger: a rendered image of RLLLLLLLRRLRR at 30M steps (visit-count colouring): thin diagonal lines forming nested 45-degree rectangles over a flat field. Reproduced exactly (examples/rulepat_dump.rs, 8192^2 board, from the repo root in WSL): ./target/release/examples/rulepat_dump RLLLLLLLRRLRR 30000000 experiments/2026-10-14-rulepat/dumps/img -> bbox 2322 x 2322, 100% of bbox visited, max visits 274 (all match the image).

Observation before any hypothesis (2026-10-14 18:50 NZST)

• 81.1% of the 5.39M visited cells have been visited exactly twice and sit at colour 2 (the flat field in the image). Visits/cell overall = 5.6.
• Area / t = 0.18 and bbox width 2322 at t = 3e7 is roughly t^0.5, i.e. area ~ t, NOT the t^(2/3) area of the "chaotic" rules this repo studies. One endpoint is not an exponent: step 2 measures it.
• 46% of steps turn opposite to the previous step (staircase / diagonal moves); ~1,500 maximal alternating runs of length 32-1023 (the rectangle sides).

Question

What property of a rule makes the ant (a) leave most cells at a single colour after a fixed small number of visits (a "plateau"), and (b) travel in long diagonal ballistic segments that bounce around inside the territory? Is it the same thing as a different growth class?

Descriptor (fixed now)

For a run of N steps: m = modal visit count among visited cells, plateau = share of visited cells with that count, pc = their colour, area/t, bbox width, long-run count (alternating-turn runs >= 32). Calibrate on controls first (step 5).

Predictions (written before steps 2-5 are run)

• P1 growth: area exponent for this rule >= 0.9 (vs 0.667 for chaotic rules), width ~ t^0.5, between-seed sd < 0.03. If the exponent is ~0.667 the plateau is just a different texture inside the known class and the story changes.
• P2 ablation (flip each of 13 letters): the pattern is set by the early letters. Flipping colour 0, 1 or 2 destroys the plateau; flipping any of colours 3..12 keeps plateau >= 0.5 in a majority (>= 6 of 10) of cases. [Rationale guess: RL prefix = Langton's ant laying a first pass 0->1, colour 2 = L then is the "parked" state; later colours only matter for the rare cells that get a third visit.]
• P3 census (data/b3_rules.txt, 1200 rules): plateau rules (plateau >= 0.5) are enriched for prefix RLL by >= 3x over base rate, and pc is 2 in most of them.
• P4 the diagonal segments are transient highways: detect.rs confirms periodic motion on some, with even period and |dx|=|dy| as in the H19 catalogue. A 2^30 run does not lock into one permanent highway (the territory keeps growing in all directions).

Log

(append below; do not rewrite the predictions above)

2026-10-14 19:20 NZST step 2 (growth class) and step 3 (ablation) done; P1 supported, P2 FALSIFIED

Commands (repo root, WSL, after cargo build --release --bin probe --example rulepat_scan): ./target/release/probe 27 16384 < results/growth_rules.txt > results/growth_probe.csv (RLLLLLLLRRLRR, @1..8 random 5x5 start patches, 2 tethered rules, 3 rotors, 2 random walks; the random walks leave the board, dropped) then uv run scripts/growth_fit.py -> results/growth_fit.txt. ./target/release/examples/rulepat_scan 26 8192 < results/ablation_rules.txt > results/ablation_scan.csv (and calib_* for the controls).

• P1 holds, with a regime change. Fit window k=21..27 (my first try, 16..26, was written before seeing the data and gave exponent 1.09 because area/t is NOT constant earlier: it dips to 0.05 at k=13-15 and climbs to ~0.19 by k=18-21; the window was moved after looking at per-octave area/t; that is a post-hoc choice, stated here). Area exponent 0.995 (sd 0.031 between 8 patch seeds), width exponent 0.498 (sd 0.016). Controls in the same fit: rotor 0.68-0.70 (t^(2/3)), the two tethered rules 0.25-0.29 (area saturating). So this rule is a diffusive, random-walk-like growth class (area ~ t, width ~ sqrt(t)); a 2D random walk has the same area/t ~ const (rw plateau ~0.17 at m=1, calib_scan.csv). It is not in the t^(2/3) chaotic class at all.
• Descriptor calibration (results/calib_scan.csv, 2^26 steps): target plateau 0.83 at m=2, colour 2; controls: RL 0.22 (highway, leaves the board), random walk 0.17-0.18 (m=1), rotor 0.02, chaotic / tethered rules 0.01-0.16. So plateau >= 0.5 is far from every control (max 0.22).
• P2 FALSIFIED. Not one of the 10 flips of colours 3..12 keeps the plateau (0/10 >= 0.5; all <= 0.25), and the flips of colours 0-2 fail too. Three outcomes appear, not two: trapped (area ~1e3, max_visits saturated: flips of letters 8-12), chaotic t^(2/3)-like (area ~3e5, plateau ~0.01: flips of letters 2-7) and the plateau phase. The family is fragile: truncating to the 11-letter prefix RLLLLLLLRRL gives plateau 0.86 (area 1.8e7!), RLLLLLLLRRLRR 0.83, RLLLLLLLRRLRRL 0.76, but RLLLLLLLRRLR, ...RRR and every other truncation/flip fail. The late letters matter for the ant's dynamics even though plateau cells never reach them (they only ever have colour <= 2), so the mechanism lives at the boundary / on the rare >=3-visit cells, not in the plateau interior.

Hypotheses for step 4 (census), written BEFORE running it (2026-10-14 19:25 NZST)

Census: every rule starting with R, lengths 2..14 (16,383 rules), 2^24 steps, 4096 board. Class "plateau-diffusive" = plateau >= 0.5 and area/t >= 0.05 and not escaped.

• C1 the class is rare: < 2% of rules overall, and absent for n <= 8.
• C2 nearly all members (>= 90%) have prefix RLL; plateau colour is 2 in >= 80%.
• C3 members have an L-run of length >= 4 right after the leading R (colours 1..4 all L) in >= 90%.
• C4 members come in extension families (like RLLLLLLLRRL, ...RRL, ...RRLRR, ...RRLRRL): if rule X is a member then at least one extension of X by 1-3 letters is also a member (checked for each member of length <= 11). If C2 fails (many members with other prefixes) the explanation is not "colour 2 is the parked state".

2026-10-14 census result (scripts/run_census.sh, census_stats.py, census_split.py; 16,382 rules, 2^24 steps, 4096 board, 1,985 escaped)

Outputs: results/census_scan.csv, census_stats.txt, census_split.txt.

• My membership rule (plateau>=0.5, area/t>=0.05, not escaped) caught 1,085 rules (6.6%), so C1 FAILED as written. Cause: it lumps two different things. 950 of them have plateau >= 0.98: regular fillings (every cell visited the same m times, m = 4, 6, 8..., area = t/m, ordered growth, no long diagonal runs). Only 135 (0.8%) are irregular plateau rules like the target (0.5 <= plateau < 0.98, random-walk-like area/t 0.05-0.43). The split at 0.98 is post hoc. None of the irregular ones is shorter than 8 letters.
• C2 FAILED: only 64/135 irregular members start with RLL, and plateau colour is 2 in 43/135 (mode visits = 2 in 41); other modes are 4, 6, 8, 10, 12 (even only). Colour 2 as "the parked state" is one sub-family (RLL...), not the explanation. C3 not met either (RLLLL prefix in 54 of 1,085 all members; not tested separately on the irregular ones). C4 supported (132/139 members of length <= 11 have a member extension within 3 letters; this includes the regular ones, so it is weak evidence).
• Caveat: classification at one horizon (2^24) is crude. The target itself only settles after k~18, so some "irregular" rules may be transients and slower ones missed. Not checked.

2^30 run (rulepat_scan 30 20000, results/long_scan.csv)

RLLLLLLLRRLRR does not lock into a highway or stop growing: 2^30 steps, bbox 15224, area/t 0.216, plateau 0.863 at m=2, 20,930 long alternating runs, no escape. Width/sqrt(t) = 0.465 (0.44 at 2^27). P4 (second half) holds.

Status / what is NOT done

• Mechanism unexplained. Established: (i) the pattern is a diffusive, random-walk-like growth class, not t^(2/3); (ii) most cells receive exactly two visits (colour 2 for this rule); (iii) it is fragile in rule space (single flips kill it), about 0.8% of R-prefixed rules up to length 14; (iv) a separate regular-filling class (5.8%) with the same area ~ t but perfectly uniform visit counts. Untested ideas: why even mode visit counts only (2,4,6,...: parity of passes needed to leave a cell back at a "parked" colour?); whether the plateau interior is entered at all after being formed (last-visit-time map); the front dynamics at the territory edge.
• P4 first half (are the diagonal segments transient highways per detect.rs, H19 period/displacement) was not run. The 45-degree rectangle sides are alternating-turn runs of 32-1023 steps (~2,000 per 2^26).
• Not yet linked from notes/; nothing committed (files are staged only).