Growth Dynamics in Equational Discovery: Power Laws and Saturation

A recent study published on arXiv (2605.23983) explores growth dynamics in deterministic equational discovery substrates within three simplified domains: arithmetic, boolean, and higher-order lists, analyzing 592 trajectories. The short-range substrate sizes conform to a power-law N(t) that is proportional to t^b, exhibiting architecture-sensitive regression (R^2 ≈ 0.82) that fails to transfer between substrates (R^2 ≈ -0.84 from arithmetic and boolean to lists). A heuristic mean-field closure model forecasts a saturating power-law dN/dt = K N^k exp(-mu N), where the pure power-law serves as a short-range approximation. Robustness checks reveal tight bootstrap intervals for (k, mu) in 4 out of 5 toy trajectories, with 1 showing degeneracy. Out-of-sample forecasting on toy data indicates a preference for pure power-law in all cases, suggesting saturation is not achieved. Two real-world growth proxies yield mixed results. The study also introduces new Mathlib/*.lean file additions.