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 = 23. Each line is isomorphic to magma#fdf14b96 (= x ◇ y = 9x + 23y mod 31, the F_31 linear magma with α = 23 ∈ {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=23) × F_31(α=23), a single F_31²-linear quasigroup with
(x₁, x₂) ◇ (y₁, y₂) = ((1 − 23)·x_i + 23·y_i mod 31)_(i=1,2)
Sibling magmas at size 961, all 4 constant-α direct products: magma#fc322ba7, magma#2b8e3c08, magma#ba90950a, plus this one (α=23).
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 · 2026-05-14 21:32:21
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-value — specifically α = 23. Each line is isomorphic to magma#fdf14b96 (= x ◇ y = 9x + 23y mod 31, the F_31 linear magma with α = 23).
Equivalently, this is the medial direct product F_31(α=23) × F_31(α=23): in F_31²-coordinates the operation is (x₁, x₂) ◇ (y₁, y₂) = ((1-23)·x_i + 23·y_i mod 31)_{i=1,2}.
Sibling magmas at size 961, all 4 constant-α direct products:
• magma#2b8e3c08 — α = 15
• magma#5f44e78d — α = 23
• magma#ba90950a — α = 27
• magma#fc322ba7 — α = 29
All 4 valid F_31 α-values (= primitive 10th roots of unity, roots of Φ_10 mod 31) are thus represented in the DB as constant-α direct products at size 961.
Display reorder coordinates each point as (i, j) = (i-th line of parallel class C_h, j-th line of class C_v), where two parallel classes are chosen, lines within each are sorted by minimum canonical label, and each point is the unique intersection. The rendered Cayley table shows the AG(2, 31) line geometry as a 31×31 block grid.
[text written by Claude]
dwrensha · 2026-05-14 23:41:03
dwrensha · 2026-05-14 21:32:21