quadratula: how much of Schröder's 990 quasigroup laws small quasigroups already witness

quadratula is a new public repository that asks how much of the implication structure of Schröder’s 990 quasigroup equational laws is already visible in small quasigroups. It enumerates every quasigroup of order 1 to 6 up to isomorphism — 1,131,984 classes, in Rust on a pinned toolchain — computes which laws each one satisfies, and measures that exhaustive floor against Bruno Le Floch’s arXiv:2603.29909: quasigroups of order at most 4 already witness 95.34% of the 726,207 law-level non-implications (91.68% of the 1,958 between the 47 classes), and going to order 6 reaches 96.70%, realising 94 of the 114 varieties and separating 42 of the 47 classes.

The floor does not finish the job, and because it is exhaustive that is a statement about the mathematics rather than about the search: 94 class-level non-implications still have no witness below order 7, so any finite counterexample to them has order at least 7. A cover of 17 quasigroups — proven optimal by ILP, against 20 for a greedy cover — witnesses everything the floor witnesses, and REPORT.md and CONTROLS.md are generated with no hand-typed numbers, with the claims re-checked independently by a separate parser and evaluator, Le Floch’s raw files, and kissat. The sections past the divider extend the exact results to orders 7–9, and nothing is certified in Lean yet.