Equation 677 Database

Size-81 (= 3^4) translation-invariant idempotent right-cancellative magma — same structural template as a1ef80ed26012f89. Carrier is F_81 viewed additively as (Z/3)^4; the operation has the form x ◇ y = x + δ(y - x) for a permutation δ : F_81 → F_81 with δ(0) = 0, where: • δ has order exactly 10 • δ^5 = -id (additive negation) • δ has cycle structure (10^8, 1) on F_81 — 8 orbits of length 10 on F_81*, fixed at 0 • δ acts as multiplication by an element of the order-10 subgroup H ⊂ F_81* on each of the 8 cosets of H; the 8 directional scalars δ_i are roots of x^4 - x^3 + x² - x + 1 mod 3 (i.e., primitive 10th roots of unity in F_81) • NOT F_3-linear → magma is non-medial Aut(magma) has order 81 × 40 = 3240; Sylow-3 = (Z/3)^4 = translation group, stabilizer of 0 is cyclic of order 40 acting F_3-linearly (= the index-2 subgroup of F_81* by multiplication). Display reorder places point with (Z/3)^4 coords (c_0, c_1, c_2, c_3) at index c_0 + 3c_1 + 9c_2 + 27c_3 with respect to a basis recovered from the Sylow-3 of Aut. Under this reorder T[a+x][a+y] = a + T[x][y] for every a in (Z/3)^4 — the nested 3×3×3×3 circulant structure is directly visible. [text written by Claude]