Equilibration horizons in embodied coevolutionary simulation: a measured account of stationarity-gate resolution

We report a set of measurements from a physically embodied predator–prey coevolutionary simulation in which morphological traits are bounded by a hard clamp. Three findings survive scrutiny: the population maximum sits pinned at the clamp under every condition tested, so no interior fitness optimum exists anywhere in the explored parameter space; the mean offset below the clamp depends on clamp position, falsifying invariance; and morphological traits reach an unbiased random walk while the neural controller continues to change directionally, measured on evolved parameters rather than on behavioural summaries. Three candidate distributional models—a clipped Gaussian, and two variants of a simulated truncation–and–clone recursion—are rejected: the first two by a pre-registered two-moment test, the third by a post-hoc out-of-sample test. We give a structural argument for why an entire class of model cannot succeed: a dimensionless single-trait model has the clamp as its only length scale and therefore forces clamp-invariance by construction, whereas the physical system measures the clamp against a fixed capture radius and terrain height. The paper’s principal methodological result is negative and, we argue, of wider use. Three separate quantitative claims from this system—a variance scaling exponent, a clamp dependence slope, and a monotonicity ordering—collapsed from quantitative to qualitative under a single root cause: equilibration horizons sized by extrapolating from transients. We measure the resolution of the standard stationarity test used to certify those horizons and find it detects drift at approximately 1 Å~ 10−3 gen−1 and is blind below 5 Å~ 10−4 gen−1, while the drifts it certified as stationary ran at 6.8 Å~ 10−5 to 1.8 Å~ 10−4 per generation— one order of magnitude below its detection threshold. We show that fitting an exponential approach to a segment of a non-monotone trajectory yields a plausible time constant and a plausible asymptote while being wrong about the direction of motion, with no warning in the fit’s own diagnostics. We recommend reporting binned trajectories and minimumbin location alongside any stationarity verdict, and treating extrapolated asymptotes as hypotheses rather than as horizon justifications.