Equational Theories Project Overview

• Equational Theories Project is an online, crowdsourced experiment in collaborative mathematics that investigates implication structures among magma identities using human insights and automated verification.
• It combines Lean-based formal proofs with automated theorem provers like Vampire and model finders to rigorously determine millions of equational implications.
• The project employs diverse proof strategies and latent space analysis, revealing large-scale algebraic and geometric structures in single-equation laws over magmas.

Searching arXiv for the most relevant ETP papers and adjacent work. {"query":"Equational Theories Project Tao Lean Vampire Krympa latent space arXiv", "max_results": 10} {"query":"(Skrzypczak, 2011) Equational theories of profinite structures (Bolan et al., 8 Dec 2025, Janota, 20 Aug 2025, Kondylidou et al., 20 May 2026, Janota et al., 18 Feb 2026, Berlioz et al., 28 Jan 2026)", "max_results": 10} The Equational Theories Project (ETP) is an online, crowdsourced experiment in “collaborative mathematics with machine assistance” centered on the implication structure of simple equational laws on magmas. Conceived in September 2024, it asked, for a canonical test-set of $4\,694$ single-equation laws in the language of a single binary operation, which of the $22\,028\,942$ nontrivial directed pairs are valid implications and which are not, both for all magmas and for finite magmas. The project reports that all $22\,028\,942$ edges of the implication graph were determined by a combination of human-generated and automated proofs, all validated by Lean; it also produced new constructions of magmas satisfying specific laws and several auxiliary results concerning finite magmas (Bolan et al., 8 Dec 2025).

1. Problem domain and formal formulation

A magma is a pair $M=(|M|,*)$ where $|M|$ is a set and $*:M\times M\to M$ is a binary operation. An equational law, or identity, is a formula of the form

$w_1(x,y,\dots)=w_2(x,y,\dots),$

where $w_1,w_2$ are formal words in the free magma generated by variables. For laws $E$ and $E'$, the project uses

$22\,028\,942$0

and also the finite-magma analogue $22\,028\,942$1, quantifying only over finite magmas. The implication graph is the directed graph $22\,028\,942$2 with $22\,028\,942$3 the chosen $22\,028\,942$4 laws and edges $22\,028\,942$5 exactly when $22\,028\,942$6; a corresponding finite graph $22\,028\,942$7 is defined using $22\,028\,942$8. Within this setup, a single equation in a magma is treated as a theory generated by one law (Bolan et al., 8 Dec 2025).

The initial benchmark fixes a single binary operation and studies universally quantified implications between identities in that operation. Typical examples include associativity implying commutativity,

$22\,028\,942$9

commutativity implying associativity,

$22\,028\,942$0

and trivializing equations such as

$22\,028\,942$1

The first two are reported to fail, while the third is valid because it forces the operation to be constant (Janota, 20 Aug 2025).

2. Collaborative architecture and formal verification

The project was organized as a repository-centered formal development. All formal definitions, proofs, and counterexamples were housed in a single GitHub repository, with each theorem tagged by a custom @[equational_result] attribute. Discussion and proof search were coordinated on Lean Zulip, while a CI pipeline automatically regenerated the list of proven and refuted implications, the Equation Explorer, Graphiti visualizations, the Finite Magma Explorer, progress dashboards, metrics, and the blueprint document linking informal proof sketches to Lean code. The “Lean blueprint” tool provided a navigable HTML document correlating informal sketches with theorem names in .lean files, allowing contributors to work on localized fragments without loading the entire development (Bolan et al., 8 Dec 2025).

A distinctive feature of the workflow was the separation between directly formalized results and closure operations. Only a small “kernel” of $22\,028\,942$2 positive entailments and $22\,028\,942$3 negative facts was directly formalized in Lean; the remainder of the $22\,028\,942$4 consequences was regenerated on the fly by transitivity, duality, and symmetry. The formal layer was built in Lean 4 with Mathlib, using a custom Magma class and a dual EquationX/LawX representation to express metatheorems about duality and reflection. ATP-generated arguments were not accepted as black boxes: proof certificates were replayed in Lean, and explicit CI checks were used to guarantee that no extra axioms entered the development (Bolan et al., 8 Dec 2025).

3. Proof methods and counterexample constructions

The project combined elementary rewriting, model search, algebraic constructions, infinitary arguments, and proof-assistant formalization. One proof stream used Lean’s simp and rw to obtain direct syntactic rewrites; this produced approximately $22\,028\,942$5 direct proofs and, after closure under transitivity, generated approximately $22\,028\,942$6 million implications. A second stream exploited duality: replacing $22\,028\,942$7 by its opposite operation $22\,028\,942$8, $22\,028\,942$9, sends $M=(|M|,*)$0 to $M=(|M|,*)$1, effectively halving the number of independent proofs needed (Bolan et al., 8 Dec 2025).

Refutations relied on a layered model-building portfolio. Exhaustive search over all magmas of sizes up to $M=(|M|,*)$2,

$M=(|M|,*)$3

tables, consumed $M=(|M|,*)$4 CPU-hours and refuted $M=(|M|,*)$5 million false implications using $M=(|M|,*)$6 small magmas. Linear and affine models of the form $M=(|M|,*)$7 over $M=(|M|,*)$8 or $M=(|M|,*)$9 refuted approximately $|M|$0 million more implications. Translation-invariant models reduced laws to functional equations on $|M|$1 in expressions such as $|M|$2 and supplied infinite refutations not visible to finite-model search. Additional methods included twisting semigroups, greedy extension constructions of countable magmas on $|M|$3, small perturbations of multiplication tables, abelian-group fiber extensions with cocycles $|M|$4, and invariant-based obstructions using variable multiplicities, subword patterns, or rewrite normal forms (Bolan et al., 8 Dec 2025).

This methodological diversity is central to the project’s mathematical character. It was not a single-tool ATP run, but a coordinated exploration in which brute force, Gröbner-style algebraic reasoning, infinitary constructions, and proof engineering were used as complementary techniques. A plausible implication is that the project’s tractability depended less on any one prover than on the existence of many structurally distinct routes to positive and negative results.

4. Automated theorem proving and certified explicit models

A major automated-reasoning component came from Vampire. One experimental report states that all $|M|$5 implications were submitted independently to five Vampire configurations, with saturation runs marking implications as proved when $|M|$6 yields the empty clause and finite-model-building runs marking them as refuted when a finite model of $|M|$7 is found. In the aggregate results, fmb 500i refuted $|M|$8 and proved $|M|$9, while satur 500i proved $*:M\times M\to M$0; subsequent longer runs solved small residual sets. The report’s headline conclusion is that Vampire can prove all considered implications that hold and can refute a vast majority of those that do not hold (Janota, 20 Aug 2025).

The project also stimulated work on infinite countermodels. In the unit equational fragment, a saturation can be read as a convergent rewrite system $*:M\times M\to M$1 defining an explicit, possibly infinite, Herbrand model $*:M\times M\to M$2. Applied to ETP benchmarks, this yielded rewrite-system certificates for cases with no finite countermodel. The study reports that Paradox/FMB found $*:M\times M\to M$3 finite counter-models, saturation alone refuted $*:M\times M\to M$4 more, $*:M\times M\to M$5 cases were “infinite-only,” $*:M\times M\to M$6 of those were provably finitely unsatisfiable via Infinox, and $*:M\times M\to M$7 remained only by saturation. The emitted rewrite systems were checked for ground confluence by CSI and for termination by TTT2, with both certificates verified by CeTA using IsaFoR; $*:M\times M\to M$8 of the $*:M\times M\to M$9 systems were fully certified in this way (Janota et al., 18 Feb 2026).

This explicit-model perspective is significant because it turns opaque saturation fixed points into executable semantic witnesses. Rather than merely showing that a negated implication is satisfiable, the method produces a rewrite system whose normal forms witness the countermodel directly. That shift from refutation to certified model extraction is one of the clearest technical byproducts of the project.

5. Mathematical outcomes and emergent large-scale structure

The completed implication graph has a nontrivial global shape. The project reports $w_1(x,y,\dots)=w_2(x,y,\dots),$0 total nontrivial pairs resolved, or $w_1(x,y,\dots)=w_2(x,y,\dots),$1 including reflexives; among them, $w_1(x,y,\dots)=w_2(x,y,\dots),$2 entailments are true ($w_1(x,y,\dots)=w_2(x,y,\dots),$3) and $w_1(x,y,\dots)=w_2(x,y,\dots),$4 are false ($w_1(x,y,\dots)=w_2(x,y,\dots),$5). The $w_1(x,y,\dots)=w_2(x,y,\dots),$6 laws collapse into $w_1(x,y,\dots)=w_2(x,y,\dots),$7 propositional-equivalence classes under mutual entailment, the largest class containing $w_1(x,y,\dots)=w_2(x,y,\dots),$8 laws, and the longest strict implication chains have length $w_1(x,y,\dots)=w_2(x,y,\dots),$9. For finite magmas, the manuscript states that all but two finite-magma implications were settled; it further identifies a lone open question, namely whether

$w_1,w_2$0

equivalently $w_1,w_2$1, holds in all finite magmas (Bolan et al., 8 Dec 2025).

A subsequent geometric analysis embeds the theories into a latent space derived from their statistical behavior on random finite magmas. In that study, the proof-theoretic preorder on approximately $w_1,w_2$2 theories contains approximately $w_1,w_2$3 million implications and approximately $w_1,w_2$4 million atomic steps; after quotienting by reversible implication, the graph has $w_1,w_2$5 vertices and $w_1,w_2$6 directed edges. In the latent space, reversible edges have mean Euclidean length approximately $w_1,w_2$7, atomic edges approximately $w_1,w_2$8, and strict edges approximately $w_1,w_2$9. The first principal component $E$0 correlates almost linearly with the theory’s expectation $E$1, with $E$2, and approximately $E$3 of quotient edges point in the positive $E$4 direction (Berlioz et al., 28 Jan 2026).

These observations suggest that the implication graph is not merely a combinatorial database. Finite-model statistics, equivalence classes, and directed proof structure appear to align in a coherent geometry, with tight reversible clusters and oriented flows toward stronger laws. The latent-space analysis does not replace proof, but it indicates that large-scale proof-theoretic organization can be detected statistically.

6. Proof minimization, interpretation, and broader context

The project also became a testbed for proof compression. In response to a challenge involving the implication $E$5, where Vampire produced a $E$6-step refutational proof, the tool Krympa converts a Vampire proof to a direct equational proof and minimizes it using brute force, heuristics, and both Vampire and the equational prover Twee. On the challenge problem, Krympa reduced the proof from $E$7 steps to $E$8 rewrite steps. On $E$9 provable ETP benchmarks, the average proof length decreased from $E'$0 to $E'$1 steps, and on the $E'$2 benchmarks with baseline length at least $E'$3, the average decreased from $E'$4 to $E'$5 steps; individual reductions included $E'$6, $E'$7, and $E'$8 (Kondylidou et al., 20 May 2026).

The project’s title uses “equational theories” in the universal-algebraic sense of laws and their implication structure, but the phrase has wider technical meanings. In model theory, a complete first-order theory is called equational when every definable set is a Boolean combination of instances of equations, a property that strengthens stability (Martin-Pizarro et al., 2017). In a topological-recogniser framework, families of recognisable sets form lattices exactly when they are definable by profinite equations (Skrzypczak, 2011). Against that broader background, ETP occupies a sharply concrete niche: it studies the implication graph generated by single magma identities, yet its methods connect universal algebra, finite model theory, automated deduction, and formal verification.

A recurring misconception is that the project was primarily an exercise in large-language-model-assisted proof generation. The published account states instead that LLMs and modern large-LLMs played only a minor role, mainly in UI code and autocompletion, while the substantive mathematical work came from human contributors, ATPs such as Vampire and Prover9, model finders such as Mace4, and Lean-based verification (Bolan et al., 8 Dec 2025). In that sense, the project is best understood as a landmark instance of machine-assisted collaboration in which human intuition, automated proof search, model construction, and proof-assistant validation were tightly interlocked.