We use Flow Lenia as a clean genotype-to-phenotype map: choose rule parameters, run the system, measure the form, then ask how the resulting morphology space bends, loops, and overlaps with 859 EmbryoMaker morphology snapshots and 232 Dryad fish body outlines.
1. The Map
The object is the genotype-to-phenotype projection. A genotype is a Flow Lenia rule parameterization. A phenotype is the measured shape and trajectory after simulation. The fiber over a phenotype is the set of parameterizations that land near the same measured form.
2. Topology
On the full Lenia cloud, the 4,096-landmark run finds 4,930 H1 intervals, with top persistence 2.78. On the no-food 8,192 cohort, exact dense TDA finds 7,275 H1 intervals, with top persistence 0.80.
The EmbryoMaker morphology snapshots have 297 exact H1 intervals with top persistence 0.29. The Dryad fish-outline cohort has 66 exact H1 intervals with top persistence 0.64. The interval counts are sample-size dependent; the comparable fact is that the same descriptor layer sees loop structure in all three clouds.
3. Transport
The strongest local result is fish-near, meaning near the Dryad outline cohort in the shared descriptor layer. A localized H1 neighborhood has persistence 0.563, Dryad-outline median distance 3.30 against a global Dryad-outline median of 5.90, and 32 matched transport groups with a stratified transport-enrichment p-value around 0.0002.
Transport is where the geometry starts acting like physics. We move through nearby parameterizations and ask whether returning to a similar phenotype also returns the same internal state. Across 4,802 flow-supported transport groups, the broad corpus has a real tail of non-trivial residue; the strongest localized signal is the fish-near loop neighborhood.
4. Interfaces and Arrangement
The polynomial-functor split is still useful. Kernels behave like interfaces: they sense neighborhood densities and contribute responses. The channel-weight matrix is arrangement: it wires those interfaces together.
