Equation 677 Database

Size-85 idempotent right-cancellative magma satisfying Eq 677 and Eq 255. The carrier is the cyclic group Z_85 (= Z_5 × Z_17 by CRT, since 85 = 5·17), and the magma is fully Z_85-translation-invariant — verified by exhibiting an order-85 fix-free magma automorphism. Structure: x ◇ y = x + f(y − x) in 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) and R_0 has cycle structure (1, 4²¹). 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. There are exactly C(85, 2) / C(5, 2) = 3570 / 10 = 357 distinct size-5 sub-magmas, and every element lies in exactly (85 − 1)/(5 − 1) = 21 of them — this is a **Steiner system S(2, 5, 85)**. (S(2, 5, v) exists for v ≡ 1 or 5 mod 20; here 85 ≡ 5 mod 20 ✓.) This is the direct size-85 analog of the 16 size-81 Family-4 magmas (magma#c5da3284, magma#8c8fe34b, etc.), which use the same Z_n + involution + S(2, 5, n) template at size 81 (where 81 ≡ 1 mod 20). The construction generalizes to any n ≡ 1 or 5 mod 20 with an appropriate involution f satisfying Eq 677; explicit DB entries currently exist at n = 81 (16 magmas, S(2, 5, 81)) and n = 85 (this one, S(2, 5, 85)). Locally on each 5-element line, f acts as F_5 negation (= the L_0 of the unique size-5 Eq 677 magma). Globally f is an involution that, restricted to each of the 357 size-5 lines through 0 (— wait, 0 is the fixed point of f and is in 21 lines, not 357 —) restricts coherently. Display reorder presents elements as 0, τ(0), τ²(0), …, τ⁸⁴(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 Cayley table is fully Z_85-translation-invariant: every row is a horizontal shift of row 0 = f, and the rendered image shows clean diagonal banding revealing the cyclic translation symmetry. [text written by Claude]