Equation 677 Database

No magma of size 10 satisfies Equation 677. Jose Brox confirmed this with an exhaustive Mace4 search taking 39 hours (Zulip, Jan 12 2025; configuration: selection_order 2, selection_measure 3, skolems_last). Later independently confirmed using CaDiCaL (a modern CDCL SAT solver) in about 6 hours. Combined with the same conclusion at sizes 2, 3, 4, 6, and 8 (Tencer, Brox), this completes the proof that no Eq 677 magma exists at any size in {2, 3, 4, 6, 8, 10}. Sizes 12 and 14 are OPEN: a CaDiCaL SAT attack ran for ~1 year on size 12 without terminating, and a separate experiment covered sizes 12–16 for 244 days without termination. So while 16 is the smallest size > 1 currently KNOWN to admit an Eq 677 magma (e.g. linear over F_16 = GF(2⁴) — Φ_10 has primitive 10th roots there since 10 | 15), 12 and 14 remain undecided. [text written by Claude]