a neural law that grows, holds, and heals

Evolution discovered a constant-size local neural update rule — eight nonzero ternary weights — whose repeated instantiation grows an exact global pattern on lattices 16× larger than anything seen in training, holds it indefinitely, and regenerates it exactly after severe damage. Experiment 4A on the MorphoHDL dynamic-systems extension. August 2026.

Setup

A quantized neural cellular automaton, kept deliberately hardware-native: every cell of a 2-D lattice holds C=6 binary state channels in registers and applies the same small threshold network synchronously — 30 inputs (self + von Neumann neighbours), 12 hidden threshold units, 6 outputs, ternary weights {−1, 0, +1}, small integer biases. The genome is the shared rule alone: 450 integers at every lattice size. Channel 0 is the scored phenotype; the rest are free latent state. Boundaries are fixed-zero (a finite organism with a physical edge — documented choice; torus is a later control). No floating point, no gradients, no coordinates, no stochastic updates.

The honest pipeline holds: each cell compiles to ordinary Morpho structure (a REG bank; each neuron a popcount tree of full adders with a ripple comparator; every recurrent path crosses REG). A vectorized stepper was verified bit-exact against the compiled circuit across grids, genomes, states and horizons before any large-lattice result was recorded, alongside nine other preregistered selftests (synchrony, neighbour indexing, boundary semantics, weight sharing, constant genome, full-channel damage, metric sanity, train/test disjointness, arbitrary-size instantiation).

Task: from a single seeded cell, develop a checkerboard whose phase is anchored at the seed — an exact, scale-independent target at every size. The primary score is phase-contrast Dice (target-phase Dice minus complement-phase Dice), so the empty lattice and the all-ones lattice both score zero — Experiment 3's lesson about trivial optima, applied in advance. Fitness is lexicographic: development at T=2N, then persistence over [2N, 4N], then recovery from 25% random deletion, then −(nonzero weights). Training sizes: 8, 12, 16 only.

Result

Zero-shot sizes were 10, 14, 20, 24, 32, 48 and 64 — never touched during evolution or selection. Every evolved law that solved training develops the exact checkerboard at every unseen size. Four of eight winners additionally hold it exactly for the entire 8N observation window and pass the full damage battery: after 10%, 25% or 40% random deletion (all channels cleared — no hidden survivors) or a 25%-area square wound, the same frozen rule regenerates the exact pattern at every size, with recovery time scaling roughly linearly in lattice diameter (~1.7N for deletions, faster for wounds). The other four winners are a second legitimate solution class: period-2 phase blinkers that grow the exact pattern and then alternate it with its complement — dynamic rather than static homeostasis.

The decoded mechanism — evolution exapts the boundary

The sparsest winner uses 8 nonzero weights — fewer than the ~20 of our hand-designed law — and works by a completely different principle. Decoded:

h4  = [ E.ch2 − E.ch0 ≥ 1 ]
h11 = [ N.ch2 + N.ch5 − S.ch0 − E.ch0 ≥ 2 ]
ch0' = [ h4 + h11 ≥ 1 ]
ch2..ch5' = constant 1        ch1' = constant 0

The rule first floods every latent channel to constant 1 — a "developed medium" flag. Interior cells then reduce to ch0' ≈ ¬(east neighbour): pure anti-alignment. What fixes the global phase is the edge of the world: cells missing neighbours see zeros where the medium flag should be, which breaks the symmetry; the pattern nucleates from a corner and anti-alignment relaxation propagates inward. The seed cell is ignored entirely. Repair is perfect for the same reason — the boundary is an indestructible phase reference, so any wound re-anchors deterministically.

Our hand law solves the same task by the textbook mechanism: a "reached" wavefront expands from the seed at speed 1, each newly reached cell latches the phase opposite to its already-reached neighbours, and damage (which clears wavefront state too) regrows from the intact surround. Two mechanistically distinct morphogenesis strategies — seed-anchored travelling wave, boundary-anchored relaxation — plus the blinker class, from one 450-int search space.

Caveats

All preregistered sizes are even; the boundary-anchored mechanism's phase consistency at odd sizes is untested (recorded, not re-run — the experiment is frozen). "Strong" means exact-for-≥2N-consecutive-steps within an 8N window, not a proof of unbounded stability. Random search also finds solutions at this task (3/8) — the checkerboard calibration is intentionally easy; evolution's advantages here are rate and the ability to then compress (the 8-weight winner). As throughout this programme, failures and successes are statements about the preregistered representation and matched budgets.

The programme so far

Next: differentiated neural tissue — can a developmental genome that places a small number of shared cell types discover functional roles (boundary cells, relays, organizers, repair cells), the neural analogue of Experiment 0's homogeneous medium plus specialised defects?

Code: PedalCore/morpho@dynamic-systems (evolve/nca_*.py, evolve/experiment4_nca.py). Related: Mordvintsev et al., Growing Neural Cellular Automata (Distill 2020), for the differentiable NCA lineage; this experiment is its quantized, evolved, hardware-native cousin. All zero-shot numbers are frozen-genome evaluations at sizes never exposed during evolution.

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.