Elimination Templates in Macaulay2

The SAIR Mathematics Distillation Challenge on Equational Theories asks a solver to classify whether one magma identity implies another and, for either verdict, to return a certificate accepted by a deterministic Lean judge. We present a single-file solver organized as a cheapest-first cascade. Its false branch combines coefficient tests over structured algebra families, bounded finite-model search, an explicit central-groupoid witness, and several infinite-carrier witnesses. Its true branch is a proof-producing ordered unit superposition procedure with Knuth-Bendix ordering, bidirectional demodulation, indexing, memoised substitution, and anytime size deepening. Search results remain outside the trusted base: successful derivations are replayed as small Lean terms, and countermodels are rechecked by the competition judge. The frozen solver is a 189,504-byte Python file with SHA-256 f2392533c9f4c03b.... In local runs through official judge revision 2848228, it produced accepted certificates for all 1,889 rows of the six public sets with no language-model calls. Separate measurements recorded full agreement on the 800 published Stage 1 evaluation-distribution problems, 100 accepted rows in the canonical Marathon manifest without tokens, and 200 accepted rows in the hosted playground. These are regression and playground measurements, not a leaderboard result and not evidence about a hidden set. All quantitative claims are tied to immutable result ledgers; the paper makes no completeness or comparative-superiority claim.

Inspired by long-standing open problems in algebraic combinatorics, we show that modern machine learning can meaningfully contribute to verifiable mathematical discoveries. In particular, we focus on the construction of simple mathematical functions under exact distributional constraints, a setting we formalize as Simple Learning Under Rigid Proportions (SLURP). We tackle this problem by introducing two methods: MapSeek-Functional, which models the desired function alternating pseudo-labeling and supervised training steps; and MapSeek-Symbolic, designed to directly produce symbolic formulas. We successfully apply both methods to a research problem in algebraic combinatorics, discovering a new combinatorial interpretation of the $q,t$-Narayana polynomials arising from representation theory. To our knowledge, this is the first such interpretation based on noncrossing partitions. Using one discovered statistic, we find a combinatorial proof of the symmetry of these polynomials in a previously unsolved case. To streamline verification and reproducibility, we release all code, including a formalization of all the mathematical discoveries of this paper in Lean 4.

Solving a continuous algebraic constraint system requires two decisions: which values satisfy the constraints, and which structural augmentation renders an unsolvable system solvable. Classical solvers answer the first well and the second only by enumeration. On that discrete decision, a candidate-conditioned repair ranker choosing among K augmentations reaches the exhaustive-search ceiling at a fraction of the calls, outperforming random (0.997 vs 0.236 balanced nonlinear menu accuracy; p < 10^-70; 0.982 +/- 0.006 across seeds) and beating a budget-matched per-candidate probe on accuracy and cost. MARC turns such a system into a factor graph, over which a graph-neural diffusion denoiser proposes assignments, descent on an exact computer-algebra energy polishes them, and an exact symbolic checker certifies solutions. Evaluations of diffusion-based proposals rarely include one control: random multi-start under the same refinement budget. Applied to our system, it sharply curtails what the learned proposal contributes on the value decision. Does it beat random multi-start at choosing satisfying assignments? Only narrowly, in a predictable regime. Across trapped low-dimensional families it ties with random restart, but dominates in high dimension, where random search fails. Once variables couple, the advantage is gone. Since all methods share one polish and one checker, best-of-K random multi-start succeeds with probability exactly 1 - (1 - q(n))^K, where q(n) is single-start reachability; one measured constant, with no free parameters, reproduces the entire curve (mean absolute error 0.012). The favorable regime is not specific to our synthetic families: across eight real-world systems in robotics, positioning, optimization, and algebra, classical multi-start solved all eight, none in the learning-favorable regime. We map the regimes in which learned proposals improve solvers.