Generalised Langton’s ants

Why does the ant
keep coming back?

Give an ant a rule for turning left or right and let it loose on a grid. It changes each square it visits, then has to live with what it wrote there.

We’re trying to understand why so many of these ants spend so long revisiting old ground. This is the research log: conjectures, experiments, and the awkward results that send us back to the code.

What we’re trying to work out ↗

Preliminary results

From the working manuscript · 9 findings

A few things the ants have taught us so far.

Measured patterns, useful predictions, and explanations that failed. Open a card’s caveats or follow the experiment to judge the evidence.

01 / GrowthMeasured

Doubling the width takes about eight times the steps.

×8as many steps

2630 blob rules have a median width exponent of 0.3326. Rotor calibration supports about ⅓, to roughly ±0.01.

Paper figure comparing finite-window growth-exponent distributions and calibration walkers.

Inspect the exponent figure ↗

How certain is the exponent?

These are finite-window measurements, mostly through 2²⁷ steps. Calibration bias matters more than the displayed decimal places; this is not an asymptotic proof for ants.

Growth census ↗

Ant RLL and rotor accumulated occupation fields at 2²² steps, shown with shared greyscale and separately fitted spatial sizes.

Ant / RLL & rotor · accumulated visits

02 / Bulk & boundaryComparison

Similar interiors hide different edges.

The accumulated visit profiles agree within 5% in the compared radial shells. The measured surface-width growth slopes differ:

0.22–0.25examined ants

0.10–0.13rotor

What the shared profile does not prove

The cone-shaped profile is a conditional consistency check. The edge comparison uses matched step counts, not matched territory sizes. These temporal slopes do not identify a spatial roughness class. In the image, each territory is fitted separately; physical sizes differ.

Profiles and shape measurements ↗

03 / Access to the edgeDiagnostic

Departures from the cube root appear in boundary occupation.

The 425 analysed logarithmic growers visit the edge increasingly rarely. The 109 faster blob rules typically show rising relative edge occupation, κ.

Six selected ant rules: edge-entry probability is comparatively steady while relative boundary occupation varies across time.

Six selected rules through 2³² steps

Accounting or explanation?

The growth-rate identity tells us which measured factor carries a deviation. It does not explain why boundary occupation changes. These six trajectories illustrate the distinction; the counts come from the wider census.

Boundary occupation ↗

04 / Opening turnsEmpirical law

Opening turns predict edge-entry cost.

RRL…

1/c ≈ 1.36 + 0.70m

The opening run of m equal turns predicts how many boundary steps a new cell costs.

Model: about 2.76 boundary steps per new cell.

0.877fresh stationary R²

0.711broader validation R²

Where the law falls short

The broader sample misses the registered R² ≥ 0.80 target. Flat-band changes to initial interior colours preserve the law, but its constants and remaining letter effects are unexplained. The slider evaluates the fitted law, not a fresh ant simulation.

Edge-entry tests ↗

05 / A conditioning trapControl

Predicting occupation does not establish a direct opening-run effect.

A prefix lookup predicts occupation in fresh stationary rules (R² = 0.633). But controlling for early area can itself create an apparent opening-run effect.

What happened in the control?

The observed conditional coefficient was 0.148. A model with no direct opening-run effect reproduced 0.145. This does not invalidate the prediction or prove the direct effect is zero. The shortening mutation’s occupation drop remains unexplained.

Conditioning and mutation tests ↗

06 / A moving referenceCalibrated

Uniform occupation has a growth-adjusted reference.

A growing territory’s uniform reference is about 1.11 in the cube-root regime. The stationary ant median is 0.814× its own reference.

Occupation / each walker’s uniform reference

Ant median

0.81×

Random walk

1.28×

Rotor

1.8×

Are these matched comparisons?

Ants are single blank-board runs across rules; walker bars are eight-seed means. Random-walk windows are earlier than the census and the growth regimes differ. The ant distribution overlaps the controls: the median is not a bound on individual rules. Turning adjustment off shows the original probe κ.

Walker-specific references ↗

07 / Looking aheadForecast

Early growth budgets predict later growth.

68.5%lower typical forecast error

Early area history estimates a growth budget that improves future-window predictions across 99 replicated rules, against a baseline from other rules.

Where forecasts fail

Rare near-stalls produce enormous relative errors. Escaped or incomplete runs are excluded. This tests persistence of a total growth budget, not independent predictive value of its separate transport factors.

Forecast design and errors ↗

Nine walkers after 65,536 steps, including ants, rotor, random walks, cookies and reinforced walks; panel spatial scales differ.

Same step count · different spatial scales

08 / Failed explanationsNegative results

The tested memory models do not reproduce the joint regime.

Independent turns, site cookies and edge reinforcement fail to reproduce both ant-like growth and boundary occupation in the tested settings.

How far does the exclusion go?

Only these tested settings are constrained. Memory can suppress exploration without yielding stable ant-like occupation. Short windows, escapes and seed variability prevent ruling out every stochastic explanation. The image shows illustrative trajectories, not ensemble exponents.

Memory-walker controls ↗

09 / A late surpriseOne trajectory

Billions of blob-like steps need not be permanent.

12.7billion steps, then a switch

RLRLLL becomes a column sweeper, laying down a repeating three-colour wake after billions of blob-like steps.

Measured late switch of RLRLLL: discovery cost changes abruptly, a period-three wake appears, and sweep rates agree with a post-hoc model.

Inspect the transition and its wake ↗

What triggered the switch?

We do not know. Two detailed checks describe one deterministic blank-board trajectory. Their agreement does not explain the onset, establish an ensemble result, or turn every board escape into a detected highway.

Follow the column sweeper ↗

Read the working manuscript ↗Finite-time results; the mechanism of cube-root growth remains open.

Results worth a look

From the experiments

RLRLLL · A late change of habit

The blob becomes a column sweeper

After roughly 12.7 billion steps on a blank grid, RLRLLL switches to sweeping up and down, laying down columns with a repeating three-colour wake. The longer the columns get, the slower it advances sideways.

One trajectory, with two detailed windows. We can describe the sweep; what triggers the switch is still unknown.

Follow the column sweeper ↗ Analysis and provenance

Opening turns · Edge entry

A short run of turns predicts the cost of fresh ground

The opening same-letter run predicts births per boundary step: 1/c ≈ 1.36 + 0.70m. It transfers to fresh stationary rules and survives changing the initial interior colours in flat-band tests.

Broader validation misses its accuracy target. The apparent effect on occupation at fixed area can arise from conditioning on area; the constants remain unexplained.

Read the mechanism tests ↗ Results, deviations and provenance

Other grids · Edge entry

The edge-entry slope falls as the ant’s loop gets longer

We moved the ant to hexagonal and triangular cells and wrote down three guesses for the slope of 1/c against the opening run before measuring. All three failed: the slope is 0.86 for a 3-cell loop, 0.70 on squares (4), and about 0.3 for 6-cell loops. Ants also sit below the uniform reference on every grid (0.74 to 0.86), so that deficit is not a feature of squares.

The falling-slope pattern is a post hoc reading of four strata; one hex fit was ruined by an outlier rule, and there is no mechanism yet.

Read the lattice test ↗ Predictions, results and provenance

Boundary use · Where the spread comes from

The first eight letters predict boundary use, but only as a family label

Across 2185 stationary rules, the first eight letters of a rule predict about half the spread in boundary occupation (held-out R² 0.62 on 162 fresh rules); the last eight and the letter counts do not. The opening run alone predicts nothing, against my own prediction. When whole six-letter openings are held out, every model falls below the training mean (lookup 0.59 to −0.16).

The signal is family membership, not a letter-level rule that transfers. The variance shares are post hoc and rough to about ±0.1; about a third of the rule-determined spread is unexplained.

Read the family hold-out ↗ Predictions, results and provenance

Tethered ants · Editing the field

A few small patches help hold an ant near home

Shuffle the colours in a small patch of an ant’s territory and usually little changes. In two tethered rules, a few patches near the centre caused much greater damage to the ant’s tendency to stay near home.

Fresh shuffles confirmed the sensitive patches. The proposed explanation involving excursion routes did not survive controlling for occupancy.

Browse the knockout maps ↗ Analysis and caveats

What we’re working on

20 October 2026

Questions guiding the work. The linked records distinguish completed tests from proposals.

Can return journeys make κ predictable?

New proposal, not yet run: measure the chances and times of crossing successive depths inside the ant’s territory, then compose them into a prediction of boundary occupation. The test must beat persistence on fresh starts and distinguish local field memory from information that only a whole rule family supplies. The first step is synthetic calibration and an independent accounting oracle.

Read the argument and explore the scaling model ↗ Prospective protocol and failure gates

A close birth cost can hide the wrong waits

Keeping both the read and written turn bits recovers a close total cost, but changes how boundary reads are spent between births. The mismatch survives counting returns in boundary reads rather than physical steps: all four compressed fields fail the native-clock audit on four pilot rules and two seeds. Interior excursion duration alone cannot explain it. Next proposal: edit one phase while preserving two turns, then track when its hidden future clock changes the path. That causal test is unrun; the main cohort remains unopened.

Local renewal experiment ↗ Native-clock failures, exact replays and seed comparisons

What would a fair comparison with other walkers show?

Each walker gets its own uniform-occupancy reference, accounting for territory growth and holes, before boundary use is compared. Observed so far: every walker we tried (memoryless, rotor, and a rotor with noise from none to full) uses its boundary more than that reference (ratio 1.26 to 2.6), while the typical ant uses it less (median 0.81, single runs). Two of our predictions about the noisy rotor failed, and why ants sit low is open.

Walker-specific references ↗ Explore noise and measurement windows ↗

When does a long-lived pattern give way?

The column sweeper is a warning about calling a rule chaotic after a long but finite run. We still need to understand its trigger, find out whether other late escapes share its mechanism, and explain what starts and stops the crystal rules’ diagonal rays.

Late escapes ↗ Crystal rays ↗

Why does the ant
keep coming back?

Nine rules, endless trails

Preliminary results