发现方程发现的生长规律受底物类型制约，有的趋于饱和，有的持续增长。
Saturating Scaling Laws for Equational Discovery: A Phenomenology of Growth Dynamics in Three Toy Substrates with Two Real-World Replications
- 用三个玩具底物和两个真实项目验证不同生长模型
- 多数情况短期符合幂律，长期则趋向饱和，但底物类型决定趋势
- 适合研究复杂系统演化、开源项目增长或模型泛化能力的学者
我们研究确定性方程发现底物中的增长动态。在三个玩具领域（算术、布尔、高阶列表；共592条轨迹）中，短程底物规模满足幂律关系 N(t) ∝ t^b，且指数 b 对架构敏感（交叉验证 R² ≈ 0.82），但跨底物不通用（算术+布尔预测列表时 R² ≈ -0.84）。启发式平均场闭合模型预测为饱和幂律形式：dN/dt = K N^k exp(-μ N)，其中纯幂律为其短程近似。三次鲁棒性检验显示：(k, μ) 的自助区间在5条轨迹中4条紧密，1条退化；对玩具数据外样本预测（前100轮拟合，预测后400轮）中，纯幂律胜出5/5，表明尚未达到饱和。在两个真实世界增长代理中结果分化：Mathlib4每月新增文件（60个月，9701个文件）支持饱和形式，其外样本预测优于纯幂律约7倍；Coq mathcomp 每月提交记录（129个月，3083次提交）则偏好纯幂律，且 μ 趋于零。动态行为在两个层面受底物条件制约：同一底物内架构到 b 的回归不可迁移，且对 N(t) 的优选函数族（纯幂律或饱和幂律）亦因底物而异。我们提出以‘带有底物条件参数 (k, μ) 的饱和幂律增长’作为工作框架，适用于已进入饱和阶段的底物。
原文摘要 · Abstract (English)
We investigate growth dynamics in deterministic equational discovery substrates. Across three toy domains (arithmetic, boolean, higher-order list; n=592 trajectories), short-range substrate sizes fit a power-law N(t) proportional to t^b. Within each substrate b is architecture-sensitive (cross-validated R^2 approximately 0.82); the regression does not transfer across substrates (arith+bool to list yields R^2 approximately -0.84). A heuristic mean-field closure model predicts a saturating power-law dN/dt = K N^k exp(-mu N) of which the pure power-law is the short-range approximation. Three robustness checks: bootstrap intervals on (k, mu) are tight in 4/5 toy trajectories and degenerate in 1/5; out-of-sample forecasting on toy data (fit first 100 epochs, predict next 400) is won by pure power-law 5/5, indicating the toy trajectories do not reach saturation; on two real-world growth proxies the result splits. New Mathlib/*.lean file additions per month (mathlib4, 60 months, 9701 files) support the saturating form on OOS forecasting by approximately 7x over pure power-law; Coq mathcomp monthly commits (129 months, 3083 commits) favour pure power-law on both tests with mu collapsing to zero. The dynamics are substrate-conditional at two levels: within-substrate architecture-to-b regressions do not transfer, and the preferred functional family for N(t) itself (pure vs. saturating power-law) differs by substrate. We propose "saturating power-law growth with substrate-conditional (k, mu), observable when the substrate has reached its saturation regime" as a working framing.
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