Size-961 = 31² idempotent right-cancellative magma satisfying Eq 677 and Eq 255 — but in a DIFFERENT family from the AG(2, 31) line construction (sibling magmas magma#2b8e3c08, magma#5f44e78d, magma#ba90950a, magma#fc322ba7 use that template).
Distinguishing features:
• <0, 1> has size 961 — 2 elements generate the whole magma, so there are no size-31 (= AG(2, 31) line) sub-magmas through 0. This rules out the AG(2, 31) line family.
• L_0 = T[0, ·] has cycle structure (1, 60¹⁶): 16 orbits of length 60 plus the fixed point at 0. So L_0 has order 60 (vs. order 10 for the AG-line family).
• The carrier is F_961 = GF(31²) additively (= (Z/31)² as elementary abelian 31²-group). Translation by 1 in this hidden structure is an order-31 fix-free magma automorphism (verified by extending {σ(0)=1, σ(1)=754}); commutativity then yields a second independent translation τ₂, giving the full (Z/31)² translation group.
Structure: x ◇ y = x + f(y − x) on (Z/31)², where f has cycle structure (1, 60¹⁶) — 16 cosets of an order-60 subgroup ⟨c⟩ in F_961* (which has order 960 = 16 · 60). On each coset f acts as multiplication by some near-field element; the 16-way coset decomposition makes this a 16-coset 'Tao Type II'-style piecewise-linear construction on F_961.
In the display-reorder coordinates, f is F_31-linear along each AXIS (f((c, 0)) = c · (8, 12) and f((0, d)) = d · (16, 8) mod 31) but NOT F_31-linear globally: f((1, 1)) = (23, 5) ≠ (24, 20) = f((1, 0)) + f((0, 1)). This axis-additivity-without-global-linearity is the structural fingerprint of NEAR-FIELD multiplication (compare magma#5ebfbb80 for the exceptional near-field of order 11², and magma#29114da6 for the exceptional near-field of order 29²). Since 31 is NOT in the Zassenhaus exceptional list {5, 7, 11, 23, 29, 59}, the near-field underlying this magma is presumably a Dickson near-field of order 31² with the order-2 Frobenius automorphism, with c of order 60 in the Dickson multiplicative group.
Display reorder presents elements as (i, j) ↦ τ₁ⁱ τ₂ʲ(0) at index 31i + j, where τ₁, τ₂ are commuting order-31 fix-free magma automorphisms (the hidden additive translations in (Z/31)²). Under this reorder the Cayley table is fully (Z/31)²-translation-invariant.
[text written by Claude]
dwrensha · 2026-05-14 21:34:37