Sim05: Lambda Calculus Chemistry

AlChemy-inspired artificial chemistry — L1 organization emergence & L2 composition failure

Experiment Summary

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L1 Runs

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L2 Tests

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L2 Coexistence

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Max Species Explored

KEY FINDING

Even with an unbounded molecule space (lambda calculus expressions are infinite), L2 composition (multi-scale) does not emerge. 0/6 pairs achieved coexistence. Unbounded space is necessary but not sufficient for multi-scale composition.

L1 Organization Formation

Each run starts with 100 random lambda expressions. Collisions (catalytic reactions A+B→A+B+C) drive the system toward a stable organization — a small set of mutually reproducing expressions.

Unique species count over collision steps. Each run converges to a small stable set (4-37 species).

L1 Run Details

Run	Seed	Final Unique Species	Species Ever Seen	Status

L2 Composition Tests

Two independent L1 organizations are combined into one simulation. Three possible outcomes:

Pair	Outcome	Survival of A	Survival of B	Final Unique

L2 Outcome Distribution

Coexistence (L2 composition) = 0%. The multi-scale composition problem persists even with unbounded space.

Cross-Simulation Comparison

Sim04 (finite: 510 species max)

Sim05 (unbounded: infinite species space)

Sim04 exhausted its finite space (510). Sim05 never exhausts — but both fail at composition.

Key Findings

THREE PATHS, SAME FAILURE

Echo (Holland's CAS model), chemical organizations (COT/Vasas), and AlChemy (lambda calculus) ALL fail at multi-scale composition — each from a different starting point (CAS theory, origin-of-life chemistry, computational theory). This convergence is strong evidence the composition problem is fundamental, not an artifact of any single approach.

References

• Fontana & Buss (1994). "The Arrival of the Fittest: Toward a Theory of Biological Organization." Bull. Math. Biol., 56, 1-64.

• Mathis, Patel, Weimer & Forrest (2024). "Self Organization in Computation & Chemistry: Return to AlChemy." arXiv:2408.12137.

• Szathmáry (1995). "A classification of replicators and lambda-calculus models of biological organization." Proc. R. Soc. B, 260(1359), 279-286.