Size-841 = 29² idempotent right-cancellative magma satisfying Eq 677 and Eq 255 (but NOT Eq 3345). Constructed using the exceptional Zassenhaus near-field of order 29².
The multiplicative group of this near-field has order 840 = 29² − 1 with structure Z₅ × Z₇ × SL(2, 3) (the binary tetrahedral group of order 24, extended by cyclic factors Z₅ and Z₇ — a non-cyclic group of order 840). The automorphism group of the magma is the sharply 2-transitive (Z/29)² ⋊ (Z₅ × Z₇ × SL(2, 3)) of order 706,440 = 841 × 840.
Operation: x ◇ y = x + f(y − x) on the additive group (Z/29)² ≅ GF(841), where f is left-multiplication by a fixed near-field element c of order 10 in the mult group. f has cycle structure (1, 10⁸⁴) — fixed at 0, with 84 orbits of length 10 partitioning F_841* (the 84 cosets of ⟨c⟩ in the order-840 multiplicative group).
f is NOT F_29-linear (so this is NOT the simple linear F_841 magma of the form x ◇ y = x + c(y − x) over GF(841) with commutative ·): in the basis used by the display reorder, f(1, 0) + f(0, 1) = (8, 23) ≠ (7, 3) = f(1, 1) in (Z/29)² coordinates. The axis-additivity of f together with its global non-additivity is the structural fingerprint of the exceptional near-field's non-distributive multiplication.
Direct analog of magma#5ebfbb80 (size 121 = 11², exceptional near-field of order 11²), which has the same template but with Z₅ × SL(2, 3) replacing Z₅ × Z₇ × SL(2, 3), and which additionally satisfies Eq 3345.
Display reorder presents elements as (i, j) ↦ τ₁^i τ₂^j(0) at index 29i + j, where τ₁, τ₂ are commuting order-29 fix-free magma automorphisms (the hidden additive translations by '1' and '2' in (Z/29)²). Under this reorder the Cayley table is fully (Z/29)²-translation-invariant; the rendered image shows diagonal banding across the 841×841 grid.
[text written by Claude]
dwrensha · 2026-05-14 20:46:15