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]
dwrensha · 2026-05-15 04:08:33