Equation 677 Database

Size-85 = 5·17 idempotent right-cancellative magma satisfying Eq 677 and Eq 255. Z_85 cyclic + involution + Steiner system S(2, 5, 85) template. Structure: x ◇ y = x + f(y − x) on the cyclic group Z_85, where f: Z_85 → Z_85 is a fixed-point-free INVOLUTION (f² = id, f(0) = 0, 42 transpositions on Z_85 \ {0}). L_0 has cycle structure (1, 2⁴²); R_0 typically (1, 4²¹). The magma is fully Z_85-translation-invariant under an order-85 fix-free magma automorphism. Sub-magma design: every pair of distinct elements generates a 5-element sub-magma isomorphic to the unique size-5 Eq 677 magma over F_5. The 357 such sub-magmas (= C(85, 2) / C(5, 2)) form a Steiner system S(2, 5, 85), with each element lying in exactly 21 sub-magmas. (S(2, 5, v) exists for v ≡ 1 or 5 mod 20; here 85 ≡ 5 mod 20 ✓.) Sibling magmas at size 85 with the same template: magma#10e5c920, magma#31ae2815, magma#4b5e27f5, magma#5d5c0411, magma#6282dd11, magma#84944e78, magma#d02731a8. They differ in which specific involution f is used. Same family as the 16 size-81 Family-4 magmas (Z_81 + involution + S(2, 5, 81), e.g. magma#c5da3284). The construction generalizes to any n ≡ 1 or 5 mod 20; explicit DB entries at small n with the same template include sizes 21, 25, 41, 61, 65, 81, 85, 125. Display reorder lays elements out as 0, τ(0), τ²(0), … along the orbit of an order-85 fix-free magma automorphism τ (= the hidden additive translation by 1 in Z_85). Under this reorder the table is fully Z_85-translation-invariant: every row is a horizontal shift of row 0 = f. [text written by Claude]