The Equational Theories Project: Advancing Collaborative Mathematical Research at Scale

We report on the Equational Theories Project (ETP), an online collaborative pilot project to explore new ways to collaborate in mathematics with machine assistance. The project successfully determined all 22 028 942 edges of the implication graph between the 4694 simplest equational laws on magmas, by a combination of human-generated and automated proofs, all validated by the formal proof assistant language Lean. As a result of this project, several new constructions of magmas satisfying specific laws were discovered, and several auxiliary questions were also addressed, such as the effect of restricting attention to finite magmas.

💡 Research Summary

The paper presents the Equational Theories Project (ETP), a large‑scale, crowd‑sourced effort to map the implication structure among the simplest equational laws for magmas. The authors selected a test set of 4,694 equations of order at most four, assigning each a unique identifier (E1–E4694). An implication E ⊢ E′ holds when every magma satisfying E also satisfies E′; this relation forms a preorder that becomes a partial order after quotienting by mutual entailment. The primary goal was to determine the full directed implication graph for these laws, both for arbitrary magmas and for the restricted case of finite magmas.

To achieve this, the project combined human‑written formalizations in the Lean proof assistant with a suite of automated theorem‑proving (ATP) tools, SAT/SMT solvers, Gröbner‑basis calculations, linear‑model searches, translation‑invariant models, and custom greedy algorithms. By exploiting symmetries (the dual operation x ⋄ y ↔ y ⋄ x) and simple rewriting rules, the team reduced the number of implications that required an explicit proof to just 10,657 (0.13 % of all true edges). The remaining true implications were generated automatically by ATPs.

For the false implications, the authors deployed a diverse toolbox. Small finite magmas were enumerated and tested by brute force; linear models of the form x ⋄ y = a x + b y provided a rich source of counterexamples; translation‑invariant models (e.g., x ⋄ y = x + f(y − x) on additive groups) reduced many cases to functional‑equation analysis; the notion of a “twisting semigroup” S_E allowed a size comparison to refute entailments; greedy construction of multiplication tables or defining functions f produced counterexamples where other methods stalled; extensions of a base magma via homomorphic projections or cohomological adjustments were also effective. Syntactic mismatches (order of terms, presence of certain subterms) were used for quick refutations.

Overall, the project formally verified 22,028,942 directed edges (22,033,636 if reflexive edges are counted) for the unrestricted implication relation, and all but two edges for the finite‑magma relation. The two unresolved cases (E677 ⊢_fin E255 and its dual) resisted all current techniques and are conjectured to be false, but a formal proof or counterexample remains elusive.

From an organizational perspective, the entire workflow was hosted on GitHub. Every Lean file was version‑controlled, and continuous integration automatically checked proofs on each commit. Over 70 contributors participated, each developing specialized proof or counterexample modules. The team built a visualization tool called Graphiti to render Hasse diagrams of the implication posets, revealing large equivalence classes (e.g., 1,496 laws equivalent to the singleton law E2). Benchmarks for AI‑assisted theorem provers were also collected, providing a baseline for future machine‑learning research in automated reasoning.

Scientific outcomes include: (1) a complete map of the implication structure among thousands of magma laws, showing that only about 37 % of all possible pairs are true implications; (2) discovery of numerous new equivalence classes and previously unknown constructions of magmas satisfying particular law combinations; (3) systematic analysis of how restricting to finite magmas changes the implication landscape, with a modest increase in true edges.

The authors argue that ETP demonstrates the feasibility of massive, formally verified collaborative mathematics. Human intuition guided the design of proof strategies, while automation handled the combinatorial explosion. They propose extending the methodology to higher‑order equations, other algebraic structures (groups, rings, quasigroups), and deeper integration of machine‑learning‑driven proof search. The paper concludes with a call for a new generation of “formal crowdsourcing” platforms that can support millions of verified statements, paving the way for ever larger collaborative discoveries in pure mathematics.