Gaia, Compressed: Pathwise and Compressor-Relative Measures of Regulation in Daisyworld

Regulation leaves a fingerprint in compression: remove the biosphere, and the planet's temperature history gets harder to describe.

The core idea is simple. A regulated system — one that holds some variable near a target despite external disturbance — should produce a more compressible output than an unregulated one. Daisyworld is the perfect test case: black and white daisies, each growing best near an optimal temperature, collectively hold their planet's surface temperature stable as the star brightens, purely through albedo feedback. No foresight, no design. The question this paper asks is: can you measure that regulation on a single trajectory, without averaging over an ensemble of planets?

The answer is yes, and the instrument is compression. The paper uses Lempel-Ziv code lengths — the same idea behind zip files — as a computable stand-in for Kolmogorov complexity, which is the theoretically ideal measure of how much information a string contains. When the biosphere is present, the planet's temperature history compresses well. When you ablate it (run the same simulation with daisies removed), the temperature history gets longer by 591 ± 109 bits, consistently, across 20 independent runs and every encoding tested. That gap is the algorithmic signature of regulation. The paper is careful about what this does and doesn't mean: it's not an estimate of the theoretical Kolmogorov gap (computable code lengths don't bound incomputable complexities), but it does carry exact evidential weight through the probability model the compressor implicitly defines. Separately, the regulated planet's surface temperature correlation with stellar luminosity drops from 0.993 to 0.582 — the biosphere is genuinely screening the planet from its star.

Two further results build on this foundation. First, the paper constructs a "viability-information curve" — borrowed from Kolchinsky and Wolpert's semantic information framework — but computed pathwise on single trajectories rather than over an ensemble. As you progressively weaken the biosphere's albedo channel, it sheds most of its ~120 bits of star-tracking information at nearly constant viability, then collapses sharply once little remains. This is a ramp, not a threshold. Standard Shannon mutual information and transfer entropy reproduce the shape of this curve well (r ≥ 0.93), but give inconsistent estimates of the "irreducible fraction" of that information (0.27, 0.47, and ~0 respectively depending on estimator) — so the curve itself is the robust result, and the fraction is an estimator artifact. Second, the paper asks whether the black-white daisy polymorphism needs to be imposed by hand. Under fast seasonal forcing, a single gray-pigment gene spontaneously splits into the standing black-white pair through disruptive selection — a branching transition driven by a storage effect when environmental excursions exceed a threshold set by generational turnover. The model's own growth law conceals the branching point under standard parameters, but a smoother growth law exposes it. Crucially, it's the standing polymorphism that holds the shared information with the star, not ongoing evolution per se.

The paper also contains a self-contained analysis of anticipation: if daisies are given a forward model of their world and allowed to act on predicted future temperatures rather than current ones, regulation improves — but only in a world with exploitable structure (seasonal forcing), and only up to a horizon near the population's own response time 1/γ. Beyond that, over-anticipation actively harms regulation by 57%. The key insight is that a regulator must forecast what it cannot influence (the exogenous forcing) and derive the rest counterfactually — a network that accurately predicts the realized future temperature (which already contains the biosphere's own response) is worse than doing nothing at the useful horizon, because it double-counts the correction the agent is about to make.