Size-961 = 31² idempotent right-cancellative magma satisfying Eq 677 and Eq 255, in the AG(2, 31) line family. The 992 = 32 × 31 size-31 sub-magmas are exactly the lines of AG(2, 31); every pair of distinct points lies on a unique line.
Construction parameter: ALL 32 parallel classes use the SAME α = 27. Each line is isomorphic to magma#543c5479 (= x ◇ y = 5x + 27y mod 31, the F_31 linear magma with α = 27 ∈ {15, 23, 27, 29} = primitive 10th roots of unity in F_31 = roots of Φ_10 mod 31).
Equivalently, this is the medial direct product F_31(α=27) × F_31(α=27), a single F_31²-linear quasigroup with
(x₁, x₂) ◇ (y₁, y₂) = ((1 − 27)·x_i + 27·y_i mod 31)_(i=1,2)
Sibling magmas at size 961, all 4 constant-α direct products: magma#fc322ba7, magma#2b8e3c08, magma#5f44e78d, plus this one (α=27).
Display reorder coordinates each point as F_31² (a, b) at index 31·a + b. F_31 addition is computable in the magma itself via the identity
x + y = T(R_0⁻¹(x), L_0⁻¹(y))
where R_0(x) = T[x, 0] (= ×(1 − α) in F_31) and L_0(x) = T[0, x] (= ×α). This lets us recover canonical → F_31² labels by BFS over the ⟨R_0, L_0⟩ action on canonical 1 (giving the horizontal axis F_31 × {0}), then on a chosen vertical generator (giving the vertical axis), then combining via the addition formula above. Under this reorder the Cayley table is fully (Z/31)²-translation-invariant AND every cell matches the medial-product formula exactly — visually a clean rainbow-diagonal pattern.
[text written by Claude]
dwrensha · 2026-05-13 12:18:05
Size-961 (= 31²) idempotent right-cancellative magma in the AG(2, 31) family: the 992 lines of the affine plane AG(2, 31) are exactly the 2-generated sub-quasigroups of this magma. Each parallel class of slope m ∈ P¹(F_31) is governed by a scalar α(m) ∈ {15, 23, 27, 29} (the primitive 10th roots of unity in F_31, = roots of Φ_5(x) = x⁴-x³+x²-x+1 mod 31); within a class the operation is x ◇ y = (1-α(m))x + α(m)y. eq677 follows from the per-line identity α^4 - α^3 + α^2 - α + 1 ≡ 0.
This seed: F_31(α=27) global (medial).
Constant α=27 on all 32 slopes. This is the medial direct product F_31(α=27) × F_31(α=27): x ◇ y = 5·x + 27·y (mod 31) componentwise. Equivalently a single linear quasigroup over F_31². Provided as the simplest AG(2, 31) seed for size 961.
[text written by Claude]
dwrensha · 2026-05-14 23:41:03
dwrensha · 2026-05-13 12:18:05