Self-Replicating
Neural Cellular Automata

Left alone, the substrate loses most of the digit. Seeded on a digit’s ink, the unmodified grid loses the shape within five generations. Its overlap with the digit is then 0.12 with random founders, and 0.01 with clonal founders once their budget runs out. With random founders, a read-out fitted to the grid’s state falls from 80.7% to 17.4% over 50 generations, still above the 11.1% chance. In the clamped frame, untrained rules mostly ignore the digit: 64 of 100 put every test digit in the same chemistry bin.

Searching the 40 weights of one shared agent (CMA-ES, three runs, biases kept at zero) gives rules that score 84.4–91.4% on the 416 test 0s and 1s, 88.9% on average. Always giving the commoner answer scores 51.9%. In pictures of run B, the “1” state starts at a stroke tip and floods along the stroke; the other runs spread it differently. Measured per run, a digit with a closed loop gets a “1” majority only 6–11% of the time, and one without a loop 75–94%. The chemistry does carry information: cells that vote from their own 3×3 view through a fitted lookup table, with no signalling, reach only 67.1%. But 0 vs 1 is an easy pair. A threshold on one number, the ink count, scores 92.8%, and “has a closed loop” alone scores 93.3%. Both beat every run.

The same search reaches only 57.2% (56.0–57.9%) on digit < 5 and 16.3% (11.4–19.75%) on ten classes, where chance is 50.2% and 11.1%. On digit < 5 the ink count alone gets 53.0%. On ten classes the nearest mean of the ink count gets 22.1%, above every run. For much of each digit < 5 search the rule gave every digit the same answer, and the rules kept come from isolated validation checkpoints, not from a learning curve. The local information is there: the cells’ 3×3 views, pooled by a trained linear read-out, give 72.5% on ten classes. But cells that vote from their own view through a fitted lookup table reach only 18.1%, so a rule has to pool evidence through its one chemical. The search found no agent with two hidden units and one chemical that does this. Freeing the four biases did not help (one seed: 57.5% and 17.6%). A wider agent was not tried.

The substrate’s own mechanism also handles 0 vs 1, with two more changes to the rule. First, agents persist on a fixed 28×28 field, and cells off the current digit lie dormant. Second, a task check replaces the budget: after 30 steps every living cell of the wrong class dies, and a neighbour refills the site by clone-and-mutate. Frozen at the end, the three seeds’ tissues scored 98.8%, 92.3% and 49.5% on the 416 test digits; the board runs the best of them, seed 1’s. Their scores swung during training: checkpoint medians were 83.4–96.6%, and they fell below 50% at 12 of 75 checkpoints. The same mechanism stays at chance on digit < 5 (49.8–50.0%) and near it on ten classes (9.5–17.9%). On 0 vs 1 the tissue has learned where 0s and 1s put their ink, not how they look. 266 of the 275 sites covered by at least ten test digits give the same vote on at least 95% of them. A fixed map of each site’s usual vote scores exactly the tissue’s 98.8%. Shuffle the agents and it answers “0” for every digit: 48.1%, below the 51.9% from always giving the commoner answer. On the board, the tissue map shows this template. Move a digit with the arrows and its cells’ votes follow the map, not the stroke, so a digit moved sideways is often misread. A digit you draw is out of distribution for every rule here, and the tissue needs it where MNIST puts its ink.

The rule is modified as on the MNIST board. The α channel is clamped to the digit as a read-only input, and here it carries the colour: red ink is clamped at α = 1.0 and green at 0.6, both above the liveness threshold. The agents write only chemistry, starting from 0, with no inheritance, mutation or budget while a digit is read. The answer is the majority of the cells’ votes (chemistry above or below 0.5) at step 30. Every cell sees the colour in its own neighbourhood without any signalling, while the digit’s identity is spread over the whole shape.

The search settles on the colour by generation 10–20 in all three runs. After that, its answers match a colour-only rule on all but at most 2 of 8,000 training digits and 1 of 2,000 test digits per run. That gives 84.8% in training, 85.5% on test digits with training colours, and 10.4% on flipped colours. The board shows this directly. Swap a digit’s colour and the answer swaps with it. Run all 200 samples and the answer equals the colour on all 200, under either colouring. The clone-and-mutate version does the same: 67.5–85.5% on training colours and 10.4–26.7% flipped. In that version, as on the MNIST board, agents persist and every cell of the wrong class dies after 30 steps. Even untrained, 10 of 100 random rules already carry the colour, read through a fitted lookup.

With colour removed (all ink at α = 1.0), the same search reaches 52.4% (51.4–53.3%). Always giving one answer scores 50.5%, and a perfect reader of the digit would score 75%. The rule is nearly constant: on the board’s 200 samples it answers “≥ 5” for 183 of them. Like the digit < 5 searches, these searches sat at a constant answer for much of their time, and the rules kept come from isolated validation checkpoints.

In the IRM paper, an MLP trained by ERM scores 87.4% in training and 17.1% on the flipped test. IRM scores 66.9% on that test, and the grayscale oracle 73.0%. The substrate fails in both directions. It takes the shortcut as ERM does, and without the shortcut it has little shape signal to fall back on.

Where every lineage also lives outside the cut, more comes back, but the refill is still not a copy. In the twelve-region mosaic 54% of the refilled cells regain their lineage, against 73% for the same disc left uncut and 25% by chance. With two lineages the healed disc is as faithful as the uncut one (84% against 83.5%). Under the preset, though, about 85% of the refilled cells carry a genome that existed nowhere before the cut, only 1.2% carry their own pre-cut genome, and cell states match barely better than chance (45% against 42%). Every refilled cell is a birth, and births are where mutation happens. The board measures the lineage and genome shares live for any cut you make; its curves marked offline are five-seed means of the centre cut.

The couplings tried between survival and a target texture did not teach the population to inpaint. Three were tried, all this page’s additions to the rule, on two stripe textures with three seeds each. With no life budget, killing every cell that is alive where the texture should be empty holds the stripes while the rule is enforced. Lift it and the grid fills in every run (6 of 6), because selection never sees whether a genome keeps empty ground empty. Killing a misplaced cell’s whole neighbourhood wiped the population out in every run, as did killing its parent under a life budget of 8. Killing the parent along with its misplaced offspring, with no budget, selected genomes for the narrow stripes in two of three seeds (the third died out) and left only about forty cells. Planted on a fresh grid with mutation off, those genomes hold the stripes; with mutation at birth back on, new mutants overgrow them again.

One genome shared by every cell can repair simple textures; the search found no rule for shapes or digits. CMA-ES searched the 40 weights of one genome (biases fixed at zero), run in every cell with ip = 1 and no mutation, so every copy is the same rule. In 12 of 12 runs it found rules that hold and repair stripes, wide stripes, a checkerboard and diagonal bands, within 24 to 432 evaluations, and those rules restore random cuts exactly in 78–100% of trials. Run the same way, between one random founder in 180 and one in 330 is already such a repairer for stripes or a checkerboard. In three runs each the search found no rule for a grid of lines, for dots, or for growing and holding a disc. On half-masked MNIST digits its best rule draws strokes straight down: 0.228 IoU on the bottom half of 300 test digits, against 0.233 for extending the last given row downwards; the other two runs scored 0.121 and 0.105. These are search failures over three runs, not proven limits of the agent. Trained neural cellular automata, with about 8,000 parameters and 16 channels per cell, regrow whole shapes, and much more reliably when damage is part of training (Mordvintsev et al., 2020); texture-trained ones repair damage without being trained to (Niklasson et al., 2021).

Put back into the substrate with mutation, the exact rules mostly break. Planted at ip = 0.2, the twelve texture rules repaired the cut in 8 of 12 runs (11 of 12 at ip = 1), and with mutation (ppp = 0.02) in 3 of 12. Every empty cell next to a living one inherits a neighbour’s genome, and may mutate it, with probability ip each generation, so mutants arise all over the intact texture; in half the runs it had degraded before the cut. For each texture, the board’s genome is the best of three at repairing held-out cuts, and these genomes survive mutation more often than the rest (12 of 40 plantings repaired, against 5 of 80), so the board’s mutation switch breaks the texture in most runs but not in all. Neither level of the board changes the rule: the first adds only a lineage label that is copied and cleared with each genome, and the second chooses only the starting grid, the genome and the settings ip and ppp. The twelve-region mosaic is this page’s own starting pattern.

It cannot be exact. Each update would have to return exactly the average of the five values it reads, and a sigmoid of two sigmoids can only bend around that line. The best fits found, which are not proven optimal, miss by 1.6 × 10⁻⁵ when r = 0.2 and by 8.8 × 10⁻³ when r = 0.9, which also takes weights above 200. The misses do not cancel out. On a flat field the agent leaves only a few temperatures unchanged, and heat drains towards the stable ones; the board computes them from each genome and marks the one that a flat field at the current mean would drain to. In one example, the board’s hot discs under the r = 0.5 genome, the mean temperature fell from 0.121 to 0.026 in 1000 steps while the exact solver kept 0.121, and the hand-built genome, whose only stable value is 0.5, warmed the same field to 0.464. Lattice gases avoid this by building conservation into the rule. This agent cannot: it never returns its input exactly over any range.

Nor did the substrate find such a genome. Left as it is, no genome in three populations beat changing nothing on a heat test. We then tied survival to the task, which changes the rule in three ways: every 20 steps three hot discs were dropped on the colony; each step, after the cells computed their outputs and before inheritance, the 5% of fully surrounded cells that missed the heat equation by most died; and the killed cells could not be copied that step, so only mutated copies of surviving neighbours could take their places. Over 20,000 steps, in three runs at r = 0.2, the population’s median test error ended at 0.30–0.51, against 1.26–1.68 with the same number of random deaths. But no genome, sampled every 500 steps, beat leaving the field unchanged (0.17) on the fixed test patches; on the colony’s own field, in one probed run, selection mostly flattened the chemistry, where the law is easy to meet. An outside search did not find one either. A blind evolution strategy (CMA-ES, one configuration, 12,000 evaluations per run), started from founder-like weights, ended at r = 0.2 with training errors of 0.08 to 0.31, against 8 × 10⁻⁵ for the built genome. Both are single setups: they show the search is hard, not that it is impossible.

What the substrate does on its own is grow. Seed a line of founders and, for most founders (18 of 20 screened at ppp 0.02, 16 of 20 without mutation), the colony advances both ways at 0.39 rows per step, behind a front that roughens as it goes. Between 50 and 625 steps the front’s width grows roughly as t^0.27 (twelve fronts from six founders chosen because their lines grew both ways; 0.14–0.36 each). An Eden-like growth rule with no networks, in which every newborn survives and which the substrate only approximates, gives the same speed and an exponent of 0.23 at the same size. Kardar, Parisi and Zhang’s theory gives 1/3 for fronts of this kind in the long run, and real growing interfaces have been measured near it, cell-colony fronts among them at 0.336. Run sixteen times longer, the network-free rule gives 0.27, or 0.31 when an intrinsic width is allowed for. That is consistent with a slow approach to 1/3, but it is not a measurement of it.

The board’s 160-wide front nears the edge after about 190 steps with a width of only about 1.8–2.4 rows, too soon to measure the exponent, so the board draws the offline curves beside the live one. The selection runs take far longer than a page can spend, so that view replays them.

Near that edge the stock fluctuates more: its coefficient of variation is 0.02 at the best rate and 0.17 at h = 0.24. Each generation also looks more like the last, with lag-1 autocorrelation rising from about 0.80 to 0.95. These are the statistical warning signs Scheffer and colleagues describe before critical transitions. No hysteresis was found here, though, so the collapse is not shown to be a tipping point of that kind. Regrowth after fishing stops also takes longer: 21 generations back to 90 % of K from the best rate, about 50 from h = 0.24. That is mostly because less is left: up to h = 0.2, a stock thinned at random to the same size regrows just as fast. Harvesting multiplies the number of distinct genomes, from 20–31 in an unfished grid to 503–590 after 3000 generations at h = 0.2 (three runs). Every removal makes room for a birth, and mutation happens only at birth. Real fish stocks can evolve under harvesting, but no tolerance to it evolved here. Stocks fished for 3000 generations are no more tolerant of fishing than fresh ones, and an engineered genome that stays alive in any neighbourhood collapses at the same harvest rates as random founders. Random genomes already sit at that ceiling, so there is nothing for selection to find. In this substrate the fishery is set by ip, the budget and the lattice, not by the agent’s weights.

As it is, the substrate has nothing to forage, and the agent cannot sense food. Give cells food that lets them skip ageing, and colonies end up on it simply because cells there live longer. When food arrives at fixed rates and is shared among a patch’s residents, richer patches hold more cells, but less than in proportion. The matching exponent is 0.72–0.81, where the input-matching rule of ideal-free theory predicts 1, so each cell on a rich patch gets more food. Real foragers often under-match too, though part of that may be a statistical artefact, and here the cause is different. Almost nothing lives between the patches (0.003–0.010 cells per site), so each patch is its own population. The exponent says how food sets a patch’s size, not where a forager chooses to go. A honeybee colony shifts its foragers towards the most profitable nectar sources; nothing here moves or recruits. On full-strength food a cell never ages, so every patch a colony reaches is held by construction. Each patch keeps its own lineage: on marginal ground, after 2000 generations, no two patches shared their most common genome in five runs. Across ground where unfed cells die out (b = 2), a colony reached a patch 3 cells away in every run within 2000 generations. It reached one 6 cells away in 9 of 10 runs, 10 cells away in 2 of 10, and 15 or 22 cells away in none.

On a double bridge, Argentine ants end up on one branch, and on the shorter one when the branches differ. Here the colony always reaches the food first by the short branch (10 of 10 runs), but then it fills both. The short branch ends with 0.343–0.344 of the branch cells against 0.344 of the branch length, and no run ends with one branch dominant. These shares count the cells sitting on each branch, not traffic along it. The chemistry channel dies with its cell, so the ground keeps no trail. An added persistent trail field and a fixed colony size, each tried at one setting, do not change the outcome: the field only makes both branches more habitable. The lattice grows by birth into every site it can live on, and nothing in it picks one route over another. A last test asks what the agent can do at best. One genome, found by a small grid search over the agent’s weights, dies when isolated: a living cell needs one living neighbour, a newborn two. It keeps 87 % of its colony on patches that cover 17 % of the grid, against 36 % for random founders. The price is exploration: it holds only two to four of the four patches, and in 3000 generations it never re-colonises one it has failed to hold.

Harvest, food (by quality or by input), walls, the trail field and the colony cap are environment rules added for these experiments. Each acts on the grid between steps, and none of them is in nca.py. The NCA update is the page’s own and is not modified. The engineered genome and the searched genome were cloned with mutation switched off. Grids are 48–96 cells wide, not the paper’s 200 × 200. The no-mortality harvest curve is the same at 64 and 96; the budgeted regimes were run at 64 only.

The rule is not what fails. Seeded with a glider, the substrate running the board’s rule genome (sigmoid, largest weight 661) misses the glider’s births at the first step and is a still block from the second. From 50% soups it holds 46, 57 and 43 cells after 200 steps, where Life holds 292, 194 and 351. These are seeds 21–23: run the board’s 50% soup to step 200, and pick it again for the next seed. The cause is inheritance. A dead cell owns no network. It must first copy one from a living neighbour, and it computes its own state only on the next step, from a neighbourhood that has already moved on. Hand every cell the genome before each step and the same genome follows Life exactly, on 7 of 7 starts for 60–200 steps. That is a change to the rule, and the board labels it as one (“Birth lag: removed”). It is offered for Life only. The other rules’ genomes leave live alpha short of 1, that value feeds back, and with them the control is not exact.

The fix lives in the second channel. The clock genome is the same rule plus chemistry weights. Live cells set their chemistry high on one step and low on the next. On a hold step every live cell stays alive and the dead cells around it inherit the genome. On a compute step every cell applies the rule to the neighbourhood that was held still, so births arrive on time. The conditions are ip = 1, ppp = 0, no lifespan limit, and every founder carrying the genome at α = chem = 1. Under them the unmodified substrate reproduces Life cell for cell at half speed: substrate step 2k equals Life step k. That held in 24 of 24 sigmoid runs and 5 of 5 tanh runs, on 32 × 32 and 64 × 64 tori, for up to 500 Life steps. The ten other rules on the board run the same way, each exact in 6 of 6 runs over 200 rule steps, against 0 of 6 without the clock. For Coral and Vote, though, three of those six runs were 15% soups that die out under the rule itself. No genome was found for HighLife (B36/S23) or Day & Night (B3678/S34678): 60 million random weight-shared genomes never produced them, and a targeted search found no design. Neither result proves that none exists, and half speed is the only exact Life found, not a proven limit.

The clock genome was designed, not evolved, and it breaks under every setting of the substrate’s own variation that was tried. Its largest weight is 661 on the board (60 for a sigmoid version with smaller output weights, 40 under tanh), against founder weights within ±1.22. On 64 × 64 soups, every ip below 1 that was tried (0.99, 0.9 and 0.5) broke it at the first Life step, in 9 of 9 runs, and every mutation rate tried (ppp from 10⁻⁴ to 0.1) broke it within 1–4 Life steps, in 12 of 12 runs. The board’s ip and ppp switches let you try those settings live. A further test used a modified rule in which a cell that gets Life wrong dies. There, clone-and-mutate did not find Life’s rule. After 5,000 generations the most common genome, in 9 of 12 soup runs, was “survive with up to three neighbours, never give birth” (447 of 512). The best genome seen scored 451, also with no births.

Trained networks do learn Life. Any cellular automaton can be written as a convolutional network, and a convolutional network can learn arbitrary rules from videos of the automaton (Gilpin). Networks with short-range connections were trained to imitate cellular automata as early as 1992 (Wulff and Hertz). Springer and Kenyon showed that the smallest convolutional network for Life rarely converges when trained, and that overcomplete networks are needed. Per step, that network has two 3×3 filters feeding a 1×1 filter: the shape of this agent, but with biases. Differentiable logic cellular automata learn Life’s rules fully and run Life in a discrete state space (Miotti and colleagues). The substrate is the reverse case: its agent can hold the rule, but its own clone-and-mutate did not find it, and its inheritance delays every birth by a step.

In that search, all 199 glider runs moved at exactly half a cell per generation along their faster axis. The census and the external search described below turned up a few slower ones, down to a quarter. Half a cell per generation along each axis is the substrate’s speed limit. A dead cell needs one generation to inherit a genome and another to compute, so nothing travels faster. The budget does the rest. A cell may live b generations and then stays dead for at least two, which makes the grid an excitable medium in the sense of Greenberg and Hastings (1978). Periodic waves that span the grid repeat every b + 2 generations (255 of 263 cases). When a founder you roll makes such waves, the board shows their period beside b + 2. The outcome classes (died out, filled, waves, chaotic, glider) are a working version of Wolfram’s (1984) classes of cellular automata.

A creature’s heading belongs to its genome, not its body. Turned by 90°, 180° or 270°, a glider never set off in the turned direction (0 of 204 turned copies), and 106 of the 107 that re-formed headed off the original way. Deleting a single live cell mostly leaves a glider: 202 of 232 deletions did, with up to 60 cells tried on each of 19 gliders. Of those 202, 69 came back exactly as before, 76 kept the same motion in a new form and 57 moved differently. A hand-written genome also glides from a single founder at every budget from 2 upwards and with no budget at all. It is a single cell at b = 2 and alternates between one and two cells at higher budgets.

Evolution is where the substrate stops. At ip = 1, every living cell replaces its genome each generation with an unmutated copy of a living neighbour’s. A newborn mutant is not yet alive, so nothing can copy it before its own genome is overwritten. Switch mutation on in the board and the count of genomes in living cells stays at one, while the newborns carry many. Offline, all 591 runs still alive after 600 generations with mutation on carried only the founder’s exact genome. Mutation still disturbs the creatures, because a newborn computes one output from its mutant genome before losing it: only 46 of those 591 runs were still bounded. Below ip = 1 mutants survive but gliders do not. At ip = 0.95 and 0.99, only 3 of 380 gliders were still moving after 600 generations, all three of them one-cell gliders, and one of the three carried a mutated genome that still glides.

Gliders do evolve readily under an external quality-diversity search, in the spirit of Leniabreeder for Lenia (Faldor & Cully, 2024). The MAP-Elites search here changes each weight as the substrate does, by up to ±1, but with probability 0.1 rather than ppp. It also varies the budget and the seed. One run of 3,000 evaluations per activation filled 61 niches (sigmoid) and 83 (tanh), 57 and 77 of them with gliders. Those gliders have periods from 2 to 40, speeds down to a quarter cell per generation and oblique headings, and one has 169 cells. The first sigmoid glider still came from a random draw. The selection is ours, not the substrate’s. In Flow-Lenia, the rule’s parameters travel with the creatures (Plantec et al., 2023). In Lenia-like rules, moving individuals that keep their integrity against obstacles were found with diversity search, a curriculum and gradient descent (Hamon et al., 2025).