Equation 677 Database

Size 85 = 5·17 currently has 7 magmas in the DB, all sharing a single structural template: **Z_85 cyclic + involution + Steiner system S(2, 5, 85)**. All are idempotent, right-cancellative, and satisfy Eq 255. Common structure: • Additive carrier: the cyclic group Z_85 (= Z_5 × Z_17 by CRT). The magma is fully Z_85-translation-invariant: there is an order-85 fix-free magma automorphism τ acting as 'translation by 1' in the hidden Z_85 structure. • Operation: x ◇ y = x + f(y − x), where f: Z_85 → Z_85 is a fixed-point-free involution (f² = id, f(0) = 0, 42 transpositions). L_0 has cycle structure (1, 2⁴²); 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 357 such sub-magmas (= C(85, 2)/C(5, 2)), each point on exactly 21, forming a **Steiner system S(2, 5, 85)**. (Necessary condition for S(2, 5, v): v ≡ 1 or 5 mod 20; here 85 ≡ 5 ✓.) Seven entries (differ in which specific involution f is chosen): • magma#31ae2815 • magma#10e5c920 • magma#5d5c0411 • magma#84944e78 • magma#d02731a8 • magma#6282dd11 • magma#4b5e27f5 Same family as the 16 size-81 Family-4 magmas (Z_81 + involution + S(2, 5, 81); see magma#c5da3284 and siblings), with 81 ≡ 1 mod 20 the next smaller admissible size. The template generalizes to any n ≡ 1 or 5 mod 20 admitting both: • A fixed-point-free involution f on Z_n satisfying Eq 677. • A Steiner system S(2, 5, n) compatible with f's transpositions inside each line. Other DB sizes where this Steiner-system + involution template is represented: 21, 25, 41, 61, 65, 81, 125. Notably size 61 has 32 such magmas (all already commented), size 81 has 16, size 41 has 8, size 65 has 9, size 25 has 29. Display reorders for all 7 size-85 entries have been set to the orbit grid (i ↦ τⁱ(0)) using the hidden Z_85 additive translation, exposing the cyclic translation symmetry as clean diagonal banding in the rendered Cayley table. [text written by Claude]