Flow Lenia · causal score sheets · 64 fresh organisms

The information landscape has folds.

Φ‑R does not locate the organism. It labels a level containing multiple history-dependent organisms—with different bodies and different causal futures.

We drove the same fresh Flow Lenia ancestors to low and high Φ‑R through two routes, matched the final score within each level, released control, and mapped the response to nine future interventions.

The result in one sentence

History is a state coordinate.

At the same Φ‑R level, changing the route moved the organism farther than moving across the entire low→high Φ‑R span.

1.15×history distance ÷ low→high score distance at release; 95% interval 1.14–1.17.

1.03×the same ratio twenty passages later. History remains as large a coordinate as score.

1.00×history displacement ÷ low→high displacement across the full causal-future cloud.

Plainly: two creatures can have nearly the same Φ‑R and still be as different as two creatures at opposite ends of the reachable Φ‑R range.

A representative four-corner slice

Same score level. Different organism.

These are the four q44 states from one strongly separated, tightly matched sheet (ancestor 27, low-ending paths). Columns differ by history; rows differ by Φ‑R level. The quantitative result comes from all 128 sheets, not from this example.

Low Phi-R, Hold history — **Low Φ‑R**Hold history

Low Phi-R, Loop history — **Low Φ‑R**Loop history

High Phi-R, Hold history — **High Φ‑R**Hold history

High Phi-R, Loop history — **High Φ‑R**Loop history

The metric geometry

The history axis is not small.

Red measures hold→loop distance while keeping Φ‑R level fixed. Teal measures low→high distance while keeping history fixed. The red axis starts larger and remains comparable through release.

At q44

Across history: 0.130
Along score: 0.114

At q64

Across history: 0.155
Along score: 0.152

Why “folds”?

The meaning of low→high depends on history.

Hellinger distance has an exact square-root Euclidean embedding, so we can compare displacement directions—not only magnitudes. At q44, the hold→loop direction agrees across low and high Φ‑R. But the low→high direction has essentially zero agreement across the two histories.

same history direction across levels: 0.242
same score direction across histories: 0.001

The history-direction agreement then decays from 0.242 at q44 to 0.048 at q64. In the long future cloud it is -0.000: the large positional memory survives, but the local directions have been remapped.

Causal future

History moves the action world, not its size.

Low Φ‑R

Matched future displacement: 0.258
Shift / repertoire width: 1.009×
Width change: 0.0017

High Φ‑R

Matched future displacement: 0.261
Shift / repertoire width: 1.015×
Width change: 0.0003

The challenge cloud is translated by roughly one full repertoire width at both levels, while its breadth barely changes. Challenge rankings and distance topology are nearly unrelated across histories. The system has not merely become “more” or “less” capable; the map from intervention to outcome has been rewritten.

One matched challenge, 900 steps later

The hidden separation becomes visible.

These four futures start from the q44 states above and receive the same c04 challenge. The different histories do not merely preserve small pixel differences; they reorganize the later body.

future from Low Phi-R, Hold history — **Low Φ‑R**Hold history · c04

future from Low Phi-R, Loop history — **Low Φ‑R**Loop history · c04

future from High Phi-R, Hold history — **High Φ‑R**Hold history · c04

future from High Phi-R, Loop history — **High Φ‑R**Loop history · c04

Matching sanity check

This is not a score-mismatch artifact.

0.9%median within-level Φ‑R mismatch across 256 matched comparisons.

0.132q44 history distance in the tightest-matched half; the full-cohort value is 0.130.

0.263future displacement in the tightest-matched half; the full-cohort value is 0.259.

Matching error was weakly negatively correlated with both q44 state distance (-0.10) and future distance (-0.07). Poor matches are not inflating the finding.

What changed conceptually

Φ‑R is a coordinate, not a summary of the organism.

organism state ≠ f(Φ‑R)
organism state ≈ f(Φ‑R, history, context)

Our earlier return loop showed that the score can come back while the organism does not. This experiment generalizes that result across the score axis. Low and high Φ‑R each contain multiple states with different intervention-response worlds. The landscape is many-to-one, history-bearing, and locally folded.

This is not merely “memory” in the everyday sense. It is a causal-geometric memory: past information-directed actions determine which futures remain reachable from the same present score.

The next direct swing

Find where the folds become organisms.

Now we can stop asking whether Φ‑R alone predicts appearance. The sharper target is the fold itself: places where nearby information levels contain sharply different bodies and futures. We should map fold strength through early development, then intervene just before the strongest fold to see whether it redirects organism emergence, fission, identity turnover, or ecological succession.

That would connect the information geometry directly back to the original Levin-inspired question: not “does one number rise?”, but “does the causal state space reorganize before a new living pattern appears?”