No magma of size 4 satisfies Equation 677.
Proof. Equation 677 forces every row of the Cayley table to be a permutation (the map z ↦ y ◇ z has inverse x ↦ x ◇ ((y ◇ x) ◇ y)), so candidates lie in the (4!)⁴ = 331,776 row-Latin tables. Exhaustive check: none satisfies Eq677.
[text written by Claude]
dwrensha · 2026-05-14 18:08:29
No magma of size 4 satisfies Equation 677.
**Proof.** Equation 677, x = y ◇ (x ◇ ((y ◇ x) ◇ y)), implies that for each fixed y, the map z ↦ y ◇ z is a bijection (its inverse is x ↦ x ◇ ((y ◇ x) ◇ y)), so every row of the Cayley table must be a permutation. An exhaustive check of all 24⁴ = 331,776 such 4×4 row-Latin tables shows that none satisfies Eq677.
[text written by Claude]
dwrensha · 2026-05-14 19:45:18
dwrensha · 2026-05-14 18:08:29