sparse defects in a traffic jam

Independent evolutionary searches repeatedly discover a quasi-uniform density classifier: a conservative traffic-rule medium plus two single-bit symmetry-breaking defects. Successful searches preserve this two-defect architecture across system sizes; failures occur predominantly as near-tie period-2 indecision rather than incorrect consensus. An experiment on the MorphoHDL dynamic-systems extension. August 2026.

Setup

We extended Google's MorphoHDL reference implementation from pure combinational functions y = f(x) to dynamical systems s[t+1] = F(s[t], x[t]): a REG state primitive, explicit feedback wires, SCC classification of cycles (synchronous vs. opt-in asynchronous), and a discrete-time simulator (tiny_morpho_seq.py). Cellular automata need no special language features — a ring CA is a register bank, two cyclic shifts, and one LUT.

Every fitness evaluation runs the entire honest pipeline:

genome  →  Morpho cell  →  compile_seq (cyclic gate graph)  →  bit-exact simulation  →  fitness

Experiment 0 genotype: one Wolfram rule per cell of a non-uniform radius-1 CA on a ring (genome length = lattice size N, by design). Tournament selection, five mutation operators, fresh training ICs each generation, disjoint held-out ICs for all reported numbers, initial densities sampled uniformly in [0,1]. Verified bit-exact against a numpy oracle throughout.

Replication: the motif is the result

Density classification (relax to all-ones iff the initial configuration has majority ones) with N=15, T=2N, 300 generations, 24 independent seeds:

Synchronization (reach the global all-0 ↔ all-1 blinking orbit) is the control contrast: 20/24 seeds solve it perfectly, but via 15 different sea rules with ~7 defects (mean sea fraction 0.52). Same fitness protocol, radically different genotype landscape: density imposes a narrow mechanism; synchronization admits many.

The mechanism, causally

Best N=15 genome: sea 226 with defects 224 @ cell 3 and 234 @ cell 10. Ablations on a fixed 5,000-IC bank:

The medium alone cannot classify (traffic conserves particle number); each defect is individually necessary; exact position is irrelevant. Sweeping all 210 ordered placements of the two defect rules shows fitness depends almost only on their directed separation (within-separation σ ≈ 0.003), peaking at separation 7 ≈ N/2 — exactly where evolution put them. Antipodal defects each process half the ring.

The defects are algebraically minimal

Sweeping all 256 rules at each defect position (everything else fixed): the evolved rule is the unique optimum at both positions, and each is exactly one truth-table bit from the sea:

Pooled across all 36 density seeds (N=15 and N=31): 78 of 88 defects are one-bit edits of their own sea rule, and the common (sea, defect) pairs are chirality-matched mirrors — (184: 248, 168) for right-moving seas, (226: 234, 224) for left-moving. Evolution repeatedly finds the same two complementary symmetry-breaking bits, in whichever transport direction it committed to first. The runner-up rule at both defect positions is 232 — the majority rule.

Where and how it fails

Strict accuracy vs. initial majority margin (20,000 ICs): perfect at margin ≥ 6.5, ≥ 0.99 at 5.5, degrading smoothly to ~0.68 only at the minimal margin (7-vs-8 ones). The failures are not slow convergence — performance is identical at T=2N, 4N and 8N. Of 1,933 failures, 92% end in a non-uniform period-2 attractor (mean ones 7.47 ≈ exact tie) and only 0.7% of all cases commit to the wrong homogeneous state. The classifier's failure mode is a hung jury, not a wrong verdict.

success: 8 ones vs 7 zeros → all-ones
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hung: 8 ones vs 7 zeros → period-2
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Space-time diagrams of the best evolved classifier (defects at cells 3 and 10). Left: particles condense into a jam that fills the ring. Right: a standing period-2 structure refuses to resolve a near-tie.

Exact census at N=15

At N=15 there are only 2¹⁵ = 32,768 initial configurations, so sampling stops being necessary. Enumerating every IC through the compiled pipeline for the best genome of each chirality:

The two independently-evolved chiral solutions — different seeds, different defect positions — have identical exact accuracy at every margin level. Performance is fully determined by the architectural law (sea chirality, two one-bit defects, directed separation 7), not by the instance. The "hung jury" is also not a diffuse phenomenon: all indecision funnels into eight specific standing period-2 waves. Note the distributional caveat: 0.771 is under the flat census, which concentrates mass at near-ties; the ~0.90 figures elsewhere use the uniform-density benchmark distribution.

Scaling

Independent evolution at each lattice size (same population, generations and IC budget; genome length = N):

At N=63 the fixed search budget bites: the seeds split into two clusters. Five never escape messy genomes (28–38 defects, strict ≤ 0.34); three find the motif — and when they do, it is the same mechanism. The best N=63 genome is sea 226 with exactly two defects, rules 224 and 234 — the identical one-bit edits found at N=15, now classifying a lattice four times larger. Its ablation reproduces the causal structure exactly (uniform sea: strict 0.03; each defect individually necessary; position irrelevant). For successful searches, defect count stays O(1) as N grows — D(N) = 2 at N = 15, 31 and 63 — with sea fraction rising to 0.97.

The frozen law

The scaling table above entangles two claims: can evolution find the motif at size N, and does the motif itself generalize. To separate them we stopped evolving entirely: instantiate the discovered recipe — transport sea + particle-absorber + hole-absorber at separation N/2 — directly at N = 15, 31, 63, 127, 255, 511 (both chiralities), and measure P(correct | initial majority excess Δ, N). Large lattices run on a numpy stepper verified bit-exact against the compiled Morpho pipeline at N ≤ 63; 2,000 ICs per (N, Δ) cell.

Three exact-looking regularities and one honest limitation:

P(correct) at T=8N vs relative majority excess Δ/N, sea 226 (linear interpolation between measured Δ). The mirror law's numbers agree within sampling noise at every cell.

First, the law transfers: the same three-rule recipe classifies at N=511 — a lattice 34× larger than anything evolution saw — with no re-tuning. For relative excess Δ/N ≥ 1/3, accuracy is essentially independent of N. The two-defect architecture is an approximately scale-invariant relative-density classifier.

Second, T=2N and T=8N results are identical at every size and margin — convergence speed scales linearly with N and extra time never rescues a hung case, now confirmed from N=15 to 511.

Third, the two chiralities agree at every (N, Δ) cell, extending the exact N=15 equivalence to all sizes tested.

The limitation: at fixed absolute excess, accuracy decays — at Δ=1 it falls from 0.65 (N=15) to 0.21 (N=511), roughly ∝ N^−1/3. Two defects absorbing a fixed excess must wait for transport to deliver it, while a near-balanced background coarsens around them; the required excess for any target accuracy grows ∝ N. Deriving that exponent from the particle/hole dynamics — and why near-balanced configurations funnel into the handful of period-2 hung states — is the open analytic question.

Demonstrated vs. not demonstrated

Demonstrated: convergence of independent searches onto the traffic-rule motif (24/24 at N=15, 11/12 at N=31); causal necessity and algebraic minimality of the two defects (unique optima among all 256 rules, one bit from the sea); translation invariance and separation-dependence of placement; exact basin structure at N=15; D(N)=2 among successful evolved solutions at N=15/31/63; and transfer of the frozen law to N=511 with approximately scale-invariant accuracy in relative excess Δ/N.

Not demonstrated: novelty relative to the full non-uniform-CA literature (a broader search is owed before claiming it); any statement about evolved solutions at N > 63; search-success scaling — the N=63 success rate (3/8) is confounded by the fixed evolutionary budget; and an analytic account of the N^−1/3 fixed-Δ decay and the period-2 hung basins (currently empirical).

General caveats: radius-1 non-uniform CAs are approximate classifiers; nothing here is an exact solution, and none of these numbers are directly comparable to historical density benchmarks (different lattice sizes, IC distributions and convergence criteria). All experiments are elementary (radius-1, binary, synchronous) — deliberately the simplest possible substrate.

What's next

Experiment 1: temporal computation under external forcing (delayed recall, temporal parity, copy-after-delay) with evolvable topology + LUTs over synchronous Boolean networks — registers break every cycle by construction. Then the question the platform was built for: replace direct genomes with recursive developmental Morpho genomes, train at small N and short horizons, and test unchanged programs at scales never seen during evolution.

Code: PedalCore/morpho@dynamic-systems (tiny_morpho_seq.py, evolve/). Upstream: Google's paradigms-of-intelligence/morpho (Apache 2.0). References: Fukś, Solution of the density classification problem with two cellular automata rules; Sipper, cellular programming; Das, Mitchell & Crutchfield, particle-based analyses of evolved CA. All reported numbers are on held-out ICs disjoint from training.

AI-generated research. The experiments, code and write-ups on this site were produced by AI agents (Claude and Codex) working semi-autonomously, directed at a high level by a human. Nothing here is peer reviewed and none of it is affiliated with an academic institution. Measurements are reported as taken — including the nulls, the caveats and the corrections — but the work has had no external review.