Which part of a rule controls access to fresh ground?
Rules with similar growth exponents still differ in time spent at the boundary. Can we predict that difference from the instructions themselves?
2026-10-kappa-rule-spaceWays into the research
Follow the experiments.
What were we trying to understand? What did we expect? What happened? Start with the story and figures, then open the methods and notebooks when you want the detail.
104 experiments
How do ants draw diagonal crystal lattices?
Some rules leave remarkably long diagonal rays. Are these self-sustaining travelling paths, repeated reversals or structures launched by a local spinner?
2026-10-crystal-rulesWhere does the lead slope live?
Does circulation frequency explain why the birth-cost slope falls with loop length?
2026-10-20-lattice-c-law-coloursDoes the edge-entry law survive a different lattice?
The square-lattice opening-run law leaves its constants unexplained. Do changing loop length and neighbour count distinguish what sets them?
2026-10-20-lattice-c-lawIs the predictive part of the rule at its start?
Is the information about kappa specific to the first letters of a rule, or would the last letters or simple letter counts do as well?
2026-10-20-kappa-spread-featuresHow much of the spread of kappa is accounted for?
Boundary occupation varies a lot between rules. How much of that variance is start-to-start noise, how much can the rule opening predict, and how much is left?
2026-10-20-kappa-spread-budgetDoes the lead alone carry the prefix signal?
Which part of the opening carries the prefix information: the lead, or the run structure of the first few runs?
2026-10-20-kappa-prefix-runsDoes the prefix signal survive unseen rule families?
The eight-letter prefix predicts held-out kappa. Is that a transferable letter-level rule, or do rules sharing an opening simply behave alike?
2026-10-20-kappa-prefix-family-holdoutIs the prefix signal additive over letters or does it need interactions?
The eight-letter lookup carries information spread over letters 2-7. Would a model that adds up one effect per letter position do as well?
2026-10-20-kappa-prefix-additiveWhich letters of the opening carry the kappa signal?
The first eight letters of a rule predict held-out kappa. Is the information in a few early letters, or spread across them?
2026-10-20-kappa-prefix-ablationCan return journeys predict boundary occupation?
Can depth-crossing dynamics explain kappa on fresh starts, larger territories and unseen rule families?
2026-10-20-kappa-first-passageWhat does a new cell cost?
Can compressed phase memory reproduce the boundary birth process, beyond its mean cost?
2026-10-20-c-law-renewal
Is the ant unusual against the right comparison?
Ant boundary occupation sits below a simple uniform reference. Do rotor and random walkers also deviate when each gets its own growth-adjusted reference?
2026-10-19-kappa-walker-reference
The opening run predicts entry, not simply occupation
Does the start of the rule control boundary time directly, or primarily the chance of stepping onto fresh ground?
2026-10-19-kappa-chiral-edge
Why does a longer opening run make fresh ground harder to enter?
The number of boundary steps needed to discover a new cell is nearly linear in the rule’s opening-run length. Where does that law come from?
2026-10-19-c-law-mechanism
What would uniform occupation actually predict?
We normalize boundary time by perimeter divided by area. Should a walker that is uniformly spread over a growing territory therefore have kappa equal to one?
2026-10-16-kappa-uniform-baselineThe square-loop explanation that failed its test
Long opening runs can make an ant circle small squares on young cells. Does that explain their apparent boundary-occupation effect?
2026-10-16-kappa-square-loopWhat information is hidden in the start of the rule?
Short prefixes predict boundary occupation surprisingly well. Which simple feature of the prefix is carrying that information?
2026-10-16-kappa-prefix-decode
A loop intervention that was deliberately not run
If short square loops keep an ant near the edge, would suppressing those loops change its boundary occupation?
2026-10-16-kappa-loop-surgeryDoes an ant’s opening instruction predict its boundary time?
A longer opening run of identical turns appears linked to boundary occupation after adjusting for area. Does it replicate on fresh rules, and does changing one letter reproduce it?
2026-10-16-kappa-leadrun-freshDoes the opening-run effect survive a different growth clock?
A rule’s initial run of identical letters predicts boundary occupation after area adjustment. Does using current rather than early area change that association?
2026-10-16-kappa-growth-clockFaster growth measurements, checked byte for byte
Can the GPU produce the same full growth and shape measurements as the CPU probe for large batches of rules and seeds?
2026-10-16-gpu-probeRunning thousands of ants without changing the answer
Can a graphics processor accelerate a large census while returning exactly the CPU’s classifications and end-state statistics?
2026-10-16-gpu-ant-sweepWhat is already known about ants and rotor walkers?
Which published results genuinely constrain our growth and boundary questions, and which similarities are only analogies?
2026-10-15-literature-reviewCan rare long absences explain boundary occupation?
An ant can spend most of its time away from the edge in a small number of long interior episodes. Is that tail budget the missing explanation?
2026-10-15-kappa-tail-budgetAre there special gates between the edge and interior?
Could a few directional crossing patterns explain why some ants spend more time at the boundary?
2026-10-15-kappa-return-gatesDo early rule letters control launches from the edge?
Changing an early letter often has a larger effect than changing a late one. Does that happen because the boundary mostly exposes early colours?
2026-10-15-kappa-prefix-exposureDoes a maturing colour field change return behaviour?
Might the age and visit history of cells explain why ants differ in returning from the interior to the edge?
2026-10-15-kappa-field-ageDoes the excursion link survive a rule-level model?
Matched pairs suggest a relationship between extreme excursions and boundary occupation. Does a split-seed model find it across whole rules?
2026-10-15-kappa-cutoff-mixed-effects
Matching size without reusing the same trajectory
Could the excursion-cutoff association be an artefact of estimating area and both compared quantities from the same ant runs?
2026-10-15-independent-areaPersistent shape or genuinely rough coastline?
The ant’s outer edge is broader than the rotor’s. Is that a slowly changing large-scale shape, or shorter-wavelength roughening?
2026-10-15-hull-checks
Long interior trips, with territory size fixed by design
Does the connection between extreme interior excursions and boundary occupation survive comparing ants at the same visited area?
2026-10-15-fixed-area-cyclesDoes the edge model beat a simple lookup table?
A model relating turning bias to fresh-ground entry looks predictive. Does it explain more than a few early rule letters already tell us?
2026-10-15-chirality-baselines
Is the split between slow and compact growers arbitrary?
The census separates very slow rules from a broad compact-growth peak. Does the headline exponent depend on exactly where we draw the cut?
2026-10-15-alpha-gapWhat do the longest trips away from the boundary tell us?
When an ant leaves the edge, how do its longest interior excursions relate to its eventual boundary occupation?
2026-10-14-t1-excursion-tomographyWhy do some rules paint a plateau with diagonal rectangles?
One rule (RLLLLLLLRRLRR) leaves most cells visited exactly twice and draws nested 45-degree rectangles. What property of a rule produces that, and is it a different growth class?
2026-10-14-rulepatDoes boundary occupation differ beyond territory size?
Boundary occupation and territory area are tied by growth. Is there a rule-specific difference left when similarly sized territories are compared?
2026-10-14-kappa-matched-area
Is the broad edge rough coastline or internal holes?
Surface cells include the rims of holes inside the territory. Could those holes explain the ant’s broader boundary compared with the rotor?
2026-10-14-hull-widthPushing the perimeter question to much longer times
Do ant perimeters approach square-root-of-area scaling, or retain a genuinely different exponent from the rotor?
2026-10-13-w4-edge-exponent-2e36
When does the cone-shaped visit profile stop working?
The accumulated cone follows under particular occupation and growth assumptions. Does it survive when noise changes a rotor’s growth law?
2026-10-13-w3
Do thin turmites have the same visit profile as compact ants?
Can the radial visit-density argument be carried over to stateful walkers with square-root-like growth?
2026-10-13-w2Does a crust at the edge control boundary occupation?
Some territories look as though they have a distinct outer layer. Does that visible structure predict how much time the ant spends near fresh ground?
2026-10-13-w1Rechecking roughness with the right size variable
Are comparisons of boundary-width exponents robust to fitting against radius rather than clock time?
2026-10-13-w-width-radius-refitCan an early growth budget predict later territory?
Does a measured transport budget remain useful when we carry it forward to a later window instead of reconstructing the same window?
2026-10-13-transport-forecast
Which rules keep an ant close to home?
Is tethering an isolated curiosity, or a recurring kind of behaviour among very slow-growing rules?
2026-10-13-tether-rules
A small home inside a growing territory
How can an ant keep returning to a small neighbourhood while its visited region continues to grow?
2026-10-13-spread-home
A field guide to the shapes ants leave behind
How different can the visited territories look when the only instruction is a repeating string of left and right turns?
2026-10-13-snapshots
Do chaotic ant territories become round?
If the growth exponent is shared, do the territories also converge to a common shape and visit-density profile?
2026-10-13-shape-limitDoes a colour class matter beyond its abundance?
Rare colours look important to tethering. Is changing those colours more damaging than changing the same number of arbitrary cells?
2026-10-13-recolourIs the rougher edge just a faster-growing radius?
The ant’s boundary width grows faster in time than the rotor’s. Does the difference survive measuring width against territory radius instead?
2026-10-13-r1-width-vs-radius
Thousands of rules, one crowded growth peak
How broadly does cube-root-like growth occur among generalised Langton’s ants, and what distinguishes slower and faster cases?
2026-10-13-probe-growth-census
The ant moves through a world it has written
How does the colour field left by an ant affect the motion of an ant started on it later?
2026-10-13-patterns-fieldsHow much does a small initial disturbance matter?
The blank-board ant is deterministic. Does a small random starting patch change its large-scale growth, or mainly the particular path?
2026-10-13-patched-starts
Where does the ant actually spend its time?
A picture of visited cells hides how unevenly the ant uses them. What do occupancy maps reveal about compact growers and tethered rules?
2026-10-13-occupancy-pages
How random is the ant inside its own territory?
A deterministic ant often looks like a random walker in the interior. Do its colours, turns and short-lag motion support that picture?
2026-10-13-mixingChecking the machinery behind the ant measurements
Can the project’s exploratory instruments measure trajectory structure and run large sweeps without changing the walk?
2026-10-13-misc-instruments
Can a simpler walker reproduce the ant’s memory?
The ant writes information into the grid and later responds to it. Which parts of its growth can simpler memory-based walkers reproduce?
2026-10-13-memory-walkers
The ants that barely enlarge their world
Why do some rules add territory so slowly while continuing to move almost all the time?
2026-10-13-log-growersReading the layers of an ant’s colour field
What does the spatial arrangement of colours reveal that an overall histogram misses?
2026-10-13-layers-coloursDoes changing the starting colour change the growth?
A rule’s letters can be arranged around a cycle. Does rotating that cycle preserve boundary occupation, or is the first colour special?
2026-10-13-k1-necklace-iccDoes a tethered ant’s visited territory keep spreading?
An ant may stay close to home most of the time while rare excursions enlarge its historical territory. Which notion of radius captures that distinction?
2026-10-13-j4-tethered-gyration
Testing one third against a calibrated comparison
Is the narrow exponent peak evidence that different chaotic rules share exactly the same growth law?
2026-10-13-item4-exponent-rotor-calibratedAre unusually fast and slow exponents reproducible?
The census contains rules far from the central growth-exponent peak. Do they remain unusual under new starting patterns?
2026-10-13-item21-slow-outliers
Are rare colours concentrated in the weak spots?
The knockout maps identify sensitive patches. Do unusual colour classes help explain why those particular patches matter?
2026-10-13-hot-tilesWaiting longer for the apparent chaos to end
How many short rules that survive an initial census later find a repeating escape path?
2026-10-13-highways-unresolved-rules-1e10-1e11Can different growth curves be put on one scale?
Rules grow territories at different rates. Can rescaling by measured boundary factors bring their growth curves together?
2026-10-13-h3-scaling-collapseIs the exponent spread really changing empty space?
Two territories can have the same width but different numbers of visited cells. Does changing filling fraction explain why their area exponents differ?
2026-10-13-h1-lacunarity-decomposition
The ant that stopped growing a blob
RLRLLL looks like a compact chaotic grower for billions of steps, then begins sweeping columns. What changes at that late transition?
2026-10-13-g1-ribbon-breakoutMaking the frontier questions measurable
Which simulation and analysis tools are needed to turn the project’s frontier questions into reproducible tests?
2026-10-13-frontier-code-patch
When does a field learn to confine its ant?
A fresh ant moves differently on an old field than on a scrambled copy. At what field age does this difference emerge?
2026-10-13-formage-ageDoes microscopic damage destroy confinement?
A deterministic trajectory can change completely after one altered cell. Must its large-scale habit of staying home change too?
2026-10-13-flip-damage
How far does the ant growth pattern travel?
Is the observed growth peculiar to one ant on a square grid, or does it recur on other lattices and in related walkers?
2026-10-13-ext-other-lattices-turmites-noise
What does it cost to grow one more cell?
Territory growth combines time spent at the boundary with the chance of entering fresh ground once there. Which part varies with the rule?
2026-10-13-edge-statisticsWhen do the important patches become important?
Are a mature tether’s sensitive regions already doing the same job early in the trajectory?
2026-10-13-e6-hot-window-ageDo other tethered ants have the same weak spots?
The first sensitive-patch maps concern only two rules. Do similar small hot regions appear in other tethered ants?
2026-10-13-e5-more-knockout-mapsDoes the manner of damage matter?
Is losing the tether determined only by how many rare-colour cells an intervention changes?
2026-10-13-e4-patch-vs-swapAre the important cells special return routes?
Do sensitive patches guide the ant home specifically, or are they simply places the ant visits frequently?
2026-10-13-e3-route-collinearityIs the sensitive region a real scale or a measuring artefact?
Does the apparent size of a tether’s important region merely reflect the radius used to define a return home?
2026-10-13-e2-sensitive-discWhy are the sensitive patches on one side of home?
A tether’s sensitive cells appear in a one-sided lobe. Is that orientation fixed by the rule or inherited from the particular colour field?
2026-10-13-e1-one-sided-hot-regionHow much of the pattern can a tether survive losing?
Tethered ants stay near home on their own mature colour fields. How gradually does that behaviour disappear as the field is scrambled?
2026-10-13-dose-corruption-doseDo several ants do more than share the work?
What changes when several ants edit the same colour field: the growth law, or mainly how much total stepping has been done?
2026-10-13-d4-many-antsHow does noise undo the ant’s slow growth?
The ant’s movement is deterministic. If we occasionally replace its prescribed turn, does its territory abruptly become random-walk-like?
2026-10-13-d3-noiseDoes the growth story extend to three dimensions?
If revisiting old territory slows a two-dimensional ant, what growth should a related walker show in three dimensions?
2026-10-13-d2-three-dimensionsWhy do some stateful ants grow a different way?
Adding an internal state produces many walkers with square-root-like area growth. Are they compact territories with a new exponent, or thin structures?
2026-10-13-d1-turmite-sqrt-clusterWhich colours occupy the interior and the edge?
Colours record visit counts modulo the rule length. Are their spatial mixtures uniform, or does the edge retain a distinct signature?
2026-10-13-colourstats
A census of the worlds ants build
What kinds of long-run behaviour emerge when we enumerate short left/right rules instead of choosing a few attractive examples?
2026-10-13-census-sweepsCan changing one square loosen the tether?
After sensitive patches have been found, does the ant’s return behaviour depend on individual cells within them?
2026-10-13-cell1
Can we trust the growth measurements?
A fitted exponent can look convincing even when the observation window is biased. How do our instruments behave on walkers with known comparison behaviour?
2026-10-13-calib-instrument-calibrationDoes the edge settle into a stable colour mixture?
The ant sees young colours near fresh ground. Does their mixture stabilize enough to support a time-independent growth description?
2026-10-13-c3-edge-colour-sharesWhat happens when the rotor’s world has a preferred direction?
The rotor walker is a comparison model with arrows stored on the grid. Does biasing those initial arrows destroy cube-root growth?
2026-10-13-c2-arrow-biasHow often does apparent chaos turn into a highway?
An ant can wander for billions of steps before finding a repeating path that carries it away. Does the chance of that late switch stay constant per decade of time?
2026-10-13-b5-late-highwaysThe missing ants were slow, not fast
Some rules were excluded because the growth fit could not be computed. Were fast-growing ants escaping the board and disappearing from the analysis?
2026-10-13-b4-unfitted-rulesDoes the cube-root pattern survive longer instructions?
The first census concentrates on short rules. Does the growth pattern persist when an ant has many more colours to cycle through?
2026-10-13-b3-long-rules
How much of the exponent spread belongs to the rule?
The census shows a distribution of growth exponents. Is that mostly a difference between rules, or variation between starting patterns?
2026-10-13-b2-between-rule-variance
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?
2026-10-13-b1-exact-exponentDoes depletion inside the territory set boundary access?
Could a shortage of visits in part of the interior explain why different rules reach fresh ground at different rates?
2026-10-13-arm-kappa-depletion-armsCan rare breakouts dominate an average?
For rule RRLLLLLLR, does changing a small starting patch eventually cease to matter, or do rare trajectories keep the growth coefficient variable?
2026-10-13-a7-rrllllllr-self-averaging
Does a tether repair damage to its own pattern?
Rare colours are unusually important in tethered territories. Are they important because the ant repeatedly repairs damage there?
2026-10-13-a6-corruption-mechanism
Can a rule’s letters tell us what its home will look like?
Many tethered ants fall into a small collection of similar home behaviours. Can a simple property of the rule label those groups?
2026-10-13-a5-home-typesWhy is the ant’s edge broader than the rotor’s?
Ant and rotor territories can grow at similar rates while their boundaries fluctuate differently. Does the difference disappear in longer runs?
2026-10-13-a4-boundary-width
Why do the visit counts form a cone?
Ants revisit the centre of their territory more than its edge. Does that radial profile reveal a special return mechanism?
2026-10-13-a3-background-cone-controlDoes the perimeter eventually behave like a smooth boundary?
A compact two-dimensional territory should have a perimeter proportional to the square root of its area. Is the measured excess a temporary effect?
2026-10-13-a2-edge-exponent
Why do some ants spend more time at the edge?
Two ants can have similar territory shapes yet spend different amounts of time near fresh ground. Can simple rule features predict that difference?
2026-10-13-a1-kappa
Which parts of an ant’s world keep it close to home?
Some ants repeatedly return to a small home region. Is that behaviour stored in particular patches of the colour pattern they have written?
2026-10-12-knockout-maps