Three foundational papers cover most of the mechanism stock for this territory: Self-Organising Textures (Niklasson 2021) for statistical losses producing manifold attractors, Particle Lenia (Mordvintsev 2022) for local-energy dynamics, and Flow-Lenia (Plantec 2023) for architectural mass conservation.

The triple intersection — statistical loss × open systems × architectural conservation, aimed at physical-style dissipative structures — sits empty.

┌──────────────────────┐  ┌──────────────────────┐  ┌──────────────────────┐
│  Self-Organising     │  │    Particle Lenia    │  │      Flow-Lenia      │
│      Textures        │  │    Mordvintsev '22   │  │      Plantec '23     │
│    Niklasson '21     │  │                      │  │                      │
└──────────┬───────────┘  └──────────┬───────────┘  └──────────┬───────────┘
           │                         │                         │
     statistical loss            local energy              architectural
    → manifold attractor         minimization               conservation
                                                       (reintegration tracking)
           │                         │                         │
           └─────────────────────────┼─────────────────────────┘
                                     │
                       ┌─────────────┴──────────────┐
                       │   this project             │
                       │   · statistical loss       │
                       │   · open systems           │
                       │   · architectural cons.    │
                       │                            │
                       │   target: physical-style   │
                       │   dissipative structures   │
                       └────────────────────────────┘

A neural cellular automaton is a tiny update rule (≈8 K parameters) applied locally to every grid cell, run for many steps, and trained end-to-end through the rollout. The interesting design surface is the loss — the kernel will learn whatever physics that loss requires it to preserve.

What makes this path different from a Houdini PDE solver is that the loss can target objects an Euler–Lagrange machinery cannot express. SGD reaches them; closed-form variational physics does not.

Conservation 005 — sand-dune baseline

The first attractor with internal time. Dunes that move.

Three-term loss (heterogeneity, anti-stasis, soft conservation) on a residual ΔM update. Twenty experiments around this configuration. Ceiling reached: a moving-dune texture. Used as the baseline against which every later loss tension is measured.

Advection-Diffusion 006 — first physical-channel pass

Horizontal bands, slow drift.

Channels are given physical identity (M for mass, V for velocity). Loss becomes an advection–diffusion tension pair. Linear advection meets quadratic Dirichlet — small ∇M is dominated by advection (structure grows), large ∇M by Dirichlet (growth caps). First time the kernel learned a non-trivial physics over named channels.

Flow-Lenia A2 — first gradient-trained SLP

A small capsule. Same shape under any seed offset.

Conservation moved from a soft penalty into the architecture (reintegration tracking transport). Same tension-pair philosophy, different substrate. Result: a translation-invariant spatially-localized pattern — top 5% of pixels hold 100% of mass; cross-position correlation 0.99+ inside the training distribution. The 'sand-dune ceiling' claim only held for ΔM-residual architectures.

Phase 8 two-field crystal — cardinal dendrite (open problem)

A four-armed cross grows out of a small seed.

A second conserved field R is added; Σ(M+R) is bit-exact. The kernel learns to convert R into M and grows a dendrite. The dendrite locks strictly to cardinal axes — an instability of sharp U fields under Mullins–Sekerka geometry, verified by self-consistent PDE substitution where the kernel drops out and the four-fold lock disappears. Open problem K1 — see §05 below.

Phase 8.5 Ginzburg-Landau labyrinth

A maze with multi-pixel walls.

Free-energy loss replaces the tension-pair. The bare A0 form collapses labyrinths to single-pixel walls. Adding a double-well term and a Dirichlet term restores a controllable interface width. The same 64² checkpoint runs unchanged at 1024² — line widths are scale-invariant, full inference at 1024² takes 71 seconds.

Phase 8.6 Coffee-Oil 256² pyramid — coherent worm

30–60 px worms flowing across the canvas.

DyNCA-style multi-scale perception: the same DoG kernel applied at scales 1, 2, 4, 8, 16, summed before the MLP. Worms grow from 5–10 px (no pyramid) to 30–60 px. Phase-separation indicator f0+f1 = 0.60. Currently the most cinematic attractor in the project.

Phase 8.6 env-as-source-sink — conditioned trajectories

Same rule, four environments, four worlds.

Environment heterogeneity is injected after every NCA step (R = relu(R + α · fBM)), with the loss left untouched. Four warm-started checkpoints diverge into four distinct attractors as α scales. The first literal evidence that the same learned rule traces different trajectories under different ambient fields. Frontier as of 2026-04-17.

• K1 — cardinal four-fold trivial. Under low-T free-energy loss, concentrated states lock to cardinal axes. Replacing perception with isotropic kernels and the Laplacian with 9-point stencils reduces the symmetry floor without closing it. Likely culprit: perception × square grid × stochastic update mask acting jointly. Rotation augmentation and hex grids are the next experiments.
• Env attractor robustness. Stage A's a2 attractor is a global attractor: inference-only fBM injection is erased within 45 steps; persistent forcing pushes the system out of distribution and it collapses. Heterogeneity has to enter at training time.
• Trajectory memory probe. The constraint-geometry argument predicts that learned kernels can encode trajectory information in their state channels. No experiment yet distinguishes a state-Markov attractor from a trajectory-conditioned one.

Three long-running directions sit downstream. Multi-scale perception is now mostly engineering — the pyramid setup landed in 8.6. Source/sink design is in active calibration via the env-injection track. The third — losses on objects that PDEs cannot express — is the line that takes the project furthest from PDE/Houdini territory. The order they ship in is loose.

┌──────────────────────────────────────────────────┐
│                                                  │
│   third diagram · content TBD                    │
│                                                  │
│   candidates                                     │
│     · project timeline (phases 0 → 8.6)          │
│     · long-term direction matrix (A / B / C)     │
│     · loss-design taxonomy                       │
│                                                  │
└──────────────────────────────────────────────────┘