About this project

This is an interactive companion to arXiv:2510.08342, Cotler, Hongler & Hudcová's “Self-replication and Computational Universality.” It is an independent project, not affiliated with the paper's authors.

Simulation fidelity

Every claim in this project falls into one of five tiers.

Exact & tested: Elementary CA engine
Rule 110 ether + glider catalog (computationally discovered, replay-tested)
Collisions
Cyclic tag systems
Quantum walk (unitarity, light cone, dispersion relation)
Cylinder walk (periodic boundary, same tested engine with wrap on)
Torus walk (2D Grover-coin walk, unitarity tested)
Sphere and Riemann projections (coordinate changes of the tested 1D and periodic walks, not new dynamics)
Life engine (B3/S23, gun emission test: population rises by exactly 20 cells over generations 30–150)
Langton loop (behavioral replication test: 1 component at t=0, ≥ 2 by t=250, ≥ 3 by t=600, strictly increasing population)
Lenia engine and frozen creature (creature discovered by this project's crescent-seed search — μ=0.13, σ=0.016 — not taken from Chan's paper; replay-tested: survives 500 steps re-seeded on a fresh field)
Life↔Lenia bridge (s=0 equality: byte-identical to the Life engine for 20 steps; s=1 matches the Lenia engine to 1e-12)
Microtubule lattice topology (seam-blinker test, hand-traced: a vertical blinker placed on the seam column bends into a horizontal phase spanning the seam with a +3 row shift after one step, then returns to its original period-2 phase after a second)
SmoothLife engine (structured-equilibrium test: from the documented seed at size 128, mean frame-to-frame change averages below 1e-3 over steps 200–300 and spatial variance exceeds 0.02 at step 300)
Plane→Riemann-sphere projection (unit-norm identity: a projected point's distance from the origin equals 1, tested to 1e-12)
Lenia garden preset (determinism: the same size and seed reproduce a byte-identical field across independent calls)
NCA weight decode (spot-value test: four decoded weights and biases from the vendored regenerating-lizard checkpoint — sha256 e4fc6cf03accac43d5f6d96fe3cce4dbef3a25100ebe82252aedc5cc25c2a8c2 — match independently verified values to six decimal places)
Flow Lenia engine (mass-conservation test: total mass drifts by less than 1e-9 from its initial value, sampled every 10th step over a 300-step run; structured-band test: by t=300, the maximum cell value lands in [1.5, 4.0] and the occupied-cell (>0.1) fraction lands in [0.05, 0.25])
Particle Lenia engine (total-energy test: at n=200, seed=5, sampled every 50 steps over a 400-step run, total energy never increases by more than the test's own 1e-6 tolerance — an empirical property of this configuration and seed, not a theorem the energy-based formulation proves in general; spacing test: mean nearest-neighbor distance lands in [0.4, 0.9] by t=400)
Evolution loop (deterministic given a seed; archive length non-decreasing over 6 generations; for the documented seed 1, mean fitness is pinned at 0.2138 at generation 0 and 0.2903 at generation 5 — endpoints only, that same run's per-generation means genuinely dip lower in between)
String neurons — this project's own 1D neural cellular automaton (our trained artifact, not a ported checkpoint: trained by this project's own build script in packages/engine/scripts, mulberry32 seed 42, 6000 iterations. Tests pin, all against those vendored weights: growth to MSE < 0.01 of the trained target by step 48; persistence within MSE < 0.015 at step 96; recovery from an 8-cell wound to MSE < 0.02 within 48 more steps. Fire rate 1.0 — every cell updates every step — and no alive-neighborhood masking are stated departures from the 2D Growing Neural Cellular Automata model)
Alfvén-wave engine (packet-speed test: a launched packet's centroid moves at v_A to within 1e-9 after 100 steps, at the tested v_A·dt; counter-packet test: two packets launched toward each other, run for 300 steps, match a single composed 300·dt step to within 1e-9 per cell; resistive single-mode decay test: a single Fourier mode's amplitude decays as exp(−ηω²t) to within a relative 1e-9 over 200 steps at the tested mode; ideal energy-conservation test: at η=0, total energy drifts by less than a relative 1e-9 over 500 steps — every one of these four bounds is scoped below the grid's Nyquist wavelength, per the engine's own documentation)
Falsification-battery demo (post-cut test: the driven candidate's energy ratio is pinned at 0.3956 after a drive cut and 300 steps, seed 1, against the self-propagating packet's 1.0000 over the same cut and steps; phase-scramble test: scrambling the packet's phases preserves its total energy to within a relative 1e-9 while its coherence collapses to 0.4737 of its pre-scramble value)
Faithful reconstruction: Non-talking-heads mechanism (head simulation, not the paper's exact rule table; one disclosed strengthening)
Neural cellular automaton port (Mordvintsev et al.'s published regenerating-lizard weights run through this project's own TypeScript inference port. Tests pin: determinism given a seed; growth to 400–1200 living cells by step 96; footprint stability, IoU > 0.85 after 32 more steps; and recovery after damage, IoU < 0.6 immediately after a wound and > 0.75 within 96 more steps. Byte-for-byte parity with the paper's own TensorFlow implementation is not claimed, and no training runs in this project — inference only, on frozen, published weights)
The /plasmata program, heredity, and battery sections (a sourced summary of the sibling Plasmata project's site, fetched 2026-08-18; Plasmata's stated Tier-0 status and its other claims are reported as of that date and are not independently verified by this project's own test suite)
Schematic: The 1D replicator animation (the paper publishes no rule table)
The Chapter 4 von Neumann plane and strip animations (VnMachineWidget) — schematics of the constructing-arm cycle drawn at the scale of the idea, not the 29-state rule table
The Atlas home's von Neumann family-card preview — a drawn schematic of the constructor-arm cycle, not a simulation
Speculation (flagged in place): Chapter 6's closing framing
Not built, and why: A 2D quantum walk on a sphere's surface, or on a hypersphere S³: a sphere admits no arbitrarily fine uniform lattice (Euler's formula forces twelve exceptional vertices at every refinement of an icosahedral mesh), and S³ compounds the same obstruction in three dimensions, so a defensible coin-and-shift rule there is an open research question, not a rendering choice. Chapter 7's sphere widget carries a 1D walk on a great circle instead.

References

Cotler, Hongler & Hudcová, “Self-replication and Computational Universality,” arXiv:2510.08342.
von Neumann, Theory of Self-Reproducing Automata, ed. Burks (1966).
Cook, “Universality in Elementary Cellular Automata,” Complex Systems 15 (2004).
Gardner, “Mathematical Games,” Scientific American 223 (1970).
Langton, “Self-reproduction in cellular automata,” Physica D 10 (1984).
Chan, “Lenia: Biology of Artificial Life,” arXiv:1812.05433 (2019).
Rafler, “Generalization of Conway's ‘Game of Life’ to a continuous domain — SmoothLife,” arXiv:1111.1567 (2011).
Hameroff & Watt, “Information processing in microtubules,” J. Theor. Biol. 98 (1982).
Mordvintsev, Randazzo, Niklasson & Levin, “Growing Neural Cellular Automata,” Distill (2020), doi:10.23915/distill.00023 — weights vendored under Apache 2.0 from distillpub/post--growing-ca.
Plantec, Hamon, Etcheverry, Oudeyer, Moulin-Frier & Chan, “Flow-Lenia: Towards open-ended evolution in cellular automata through mass conservation and parameter localization,” arXiv:2212.07906 (2022).
Mordvintsev, Randazzo & Niklasson, “Particle Lenia and the energy-based formulation” (2022, google-research/self-organising-systems).
Lehman & Stanley, “Abandoning objectives: evolution through the search for novelty alone,” Evolutionary Computation 19(2) (2011).
Bedau et al., “Open problems in artificial life,” Artificial Life 6(4) (2000).
Meyer, “From Quantum Cellular Automata to Quantum Lattice Gases,” J. Stat. Phys. 85 (1996).
Bialynicki-Birula, “Weyl, Dirac, and Maxwell equations on a lattice,” Phys. Rev. D 49 (1994).
Thorn, string-bit models — “Reformulating String Theory with the 1/N Expansion” (1991).
't Hooft, The Cellular Automaton Interpretation of Quantum Mechanics, Springer (2016).

Related projects

Plasmata — a search for hereditary attractors in plasma dynamics: the same question this atlas asks of lattices, posed to ionized matter. Full page: /plasmata →

Source code and test suite are on GitHub. Every claim this project makes about the paper is measured against it, not copied from it.