Saturating Scaling Laws for Equational Discovery: A Phenomenology of Growth Dynamics in Three Toy Substrates with Two Real-World Replications

Computer Science > Artificial Intelligence [Submitted on 14 May 2026] Saturating Scaling Laws for Equational Discovery: A Phenomenology of Growth Dynamics in Three Toy Substrates with Two Real-World Replications Fabio Rovai 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. Comments: 17 pages, 5 figures, 4 tables, 2 algorithms. Code and data at this https URL (currently private; will be made public on acceptance) Subjects: Artificial Intelligence (cs.AI); Logic in Computer Science (cs.LO); Social and Information Networks (cs.SI) MSC classes: 68T05, 68Q32 ACM classes: I.2.6; F.4.1 Cite as: arXiv:2605.23983 [cs.AI]   (or arXiv:2605.23983v1 [cs.AI] for this version)   https://doi.org/10.48550/arXiv.2605.23983 Focus to learn more Submission history From: Fabio Rovai [view email] [v1] Thu, 14 May 2026 21:37:29 UTC (684 KB) Access Paper: HTML (experimental) view license Current browse context: cs.AI < prev   |   next > new | recent | 2026-05 Change to browse by: cs cs.LO cs.SI References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)