Saturating Scaling Laws for Equational Discovery: A Phenomenology of Growth Dynamics in Three Toy Substrates with Two Real-World Replications
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917526151102464 |
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| author | Rovai, Fabio |
| author_facet | Rovai, Fabio |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_23983 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Saturating Scaling Laws for Equational Discovery: A Phenomenology of Growth Dynamics in Three Toy Substrates with Two Real-World Replications Rovai, Fabio Artificial Intelligence Logic in Computer Science Social and Information Networks 68T05, 68Q32 I.2.6; F.4.1 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. |
| title | Saturating Scaling Laws for Equational Discovery: A Phenomenology of Growth Dynamics in Three Toy Substrates with Two Real-World Replications |
| topic | Artificial Intelligence Logic in Computer Science Social and Information Networks 68T05, 68Q32 I.2.6; F.4.1 |
| url | https://arxiv.org/abs/2605.23983 |