langtons_highway: four follow-up experiments, two of which correct earlier claims

[CrazyDubya]

What this is

Four experiments that probe the Langton's Ant highway conjecture from angles the brute-force survey (#14) and the SAT catalogue (#17) cannot reach. Two of them correct claims this README previously made, and a review round corrected a third.

1. Is Langton's rule special? No. (turmite_core.c, turmite_survey.py)

The ant and its highway certificate are generalised to any n-colour turmite rule (a string of L/R; RL is Langton's). The certificate carries over unchanged with "white" becoming "colour 0", because a cell's colour is still its visit count mod n.

All 4,095 rules of length 1–12, 20M steps each:

outcome	rules
certified highway	468	11.4%
no highway within budget	3,365
grew off the grid	250
proved periodic	12

The highway fraction holds near 11% at every rule length from 6 to 12. There are 135 distinct periods, longest 1,968; period 104 is shared by only 11 rules and the most common is 18. 65% of highway rules reach their highway faster than Langton's 9,977 steps, the quickest (RRL) in 39. Drifts are usually diagonal but not always: (16,0), (8,0), (0,2), (12,−8), (−3,11) all occur.

Soundness fix along the way: an initial "stopped visiting new cells" heuristic wrongly called LRRRRRLLR bounded after it had densely filled a 1468×1468 region and was still growing. Bounded behaviour is now proved via Brent cycle detection over an incremental grid hash.

Budget caveat, tested rather than asserted. The largest exactly-measured onset is 18,962,907 against a 20,000,000 budget, so the cutoff is inside the onset distribution. I reran 150 randomly chosen timed-out rules at ten times the budget: none converted. The figure looks stable, but it is a sample and remains an upper bound in principle.

Six of the 468 onsets are bounds rather than measurements, each stopping exactly at the ring-buffer wall. They are flagged per-entry with onset_exact and named under onset_inexact_rules, and never counted toward the exact maximum.

2. The highway cannot be destroyed at this scale (trap.py)

Obstacles dropped in the road of a running highway. Of 245 trials, 188 actually struck:

recovered a certified highway	188 of 188
never recovered	0
recovery after contact	311 min / 7,047 median / 56,117 max
period of the new highway	104, every time
same direction / deflected	67 / 121

A 21-cell wall across the road is absorbed in a few thousand steps. The highway is easy to deflect and impossible to destroy here, the dynamical counterpart of #17's uniqueness result.

Soundness detail: the certificate treats everything outside the visited bounding box as background, so obstacles are placed only after widening that box. Otherwise it would certify a highway about to hit a wall. There is a test for this.

3. The past is the future rotated 180° (backward.py)

The forward map is a bijection, so the ant has a unique past. Iterating the inverse step from an empty grid yields no new system:

backward position at step n == rot180(forward position at step n-1) + (0,-1)

Zero mismatches over 50,000 steps, grids agreeing at every checkpoint. Same onset 9,977, same period 104, drift negated. So the conjecture is time-symmetric, and reversibility independently rules out any proof that argues by shrinking basins — distinct configurations never merge, and the highway is not an attractor in the contracting sense.

4. Correction: the maximum onset does NOT grow with box size

The exhaustive maxima (43,264 / 119,673 / 233,232 for 3×3, 4×4, 5×5) drew 512, 65,536 and 33,554,432 patterns. They differ in sample size as much as in box size. At an equal 50,000 samples each:

box	max onset	mean	median	99.9th pct
4×4	119,673	3,795	1,732	47,476
5×5	88,940	4,253	2,042	57,877
6×6	112,767	4,520	2,284	59,569
7×7	107,981	4,745	2,499	58,872
8×8	126,826	4,984	2,677	57,985

The maximum is flat. Only the typical onset grows, and mildly (median +55% from 4×4 to 8×8). The earlier reading was a sample-size artifact. All 250,000 samples reached a certified highway, extending the evidence to 6×6, 7×7 and 8×8, never covered exhaustively.

Review round

Three P2 findings from the Codex bot, all correct, all mine, fixed in ec3fe27 with a regression test each:

1. The summary dropped the core's onset_exact flag, presenting six truncation bounds as measurements. The largest of them, 19,171,329, had reached the README; it now quotes the largest exact onset instead.
2. cells_per_period was actually the cumulative visited count (RRL: period 18, value 286). Renamed to visited_at_certification.
3. backward.py's onset scan started past the end of its array, so it always printed the run length instead of the onset. It now recovers 9,977, as the time-reversal symmetry requires.

summarize() was split out so --summarize-only rebuilds the summary from an existing rules.csv without repeating the sweep.

Testing

pytest langtons_highway/tests -o addopts="" → 14 passed. flake8 . --select=E9,F63,F7,F82 → 0.

🤖 Generated with Claude Code

https://claude.ai/code/session_01DVvB1dWEhjyBKh95Zojt3F

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