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 = 29. Each line is isomorphic to magma#7ee5b4b8 (= x ◇ y = 3x + 29y mod 31, the F_31 linear magma with α = 29 ∈ {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=29) × F_31(α=29), a single F_31²-linear quasigroup with
(x₁, x₂) ◇ (y₁, y₂) = ((1 − 29)·x_i + 29·y_i mod 31)_(i=1,2)
Sibling magmas at size 961, all 4 constant-α direct products: magma#2b8e3c08, magma#5f44e78d, magma#ba90950a, plus this one (α=29).
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 21:23:46
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 the affine plane AG(2, 31): every pair of distinct points lies on a unique line, and each line is itself the unique-up-to-iso linear F_31 magma associated with that parallel-class's α-parameter.
Construction parameter (identified by canonicalizing one line's 31×31 sub-table and matching it against the DB's size-31 entries): ALL 32 parallel classes use the SAME α-value — specifically α = 29. So each line is isomorphic to magma#7ee5b4b8 (= x ◇ y = 3x + 29y mod 31, the F_31 linear magma with α = 29 ∈ {15, 23, 27, 29} = the four primitive 10th roots of unity in F_31, which are the roots of Φ_10(α) = α⁴ − α³ + α² − α + 1 mod 31).
Equivalently, this magma is the medial direct product F_31(α=29) × F_31(α=29): in F_31²-coordinates the operation is
(x₁, x₂) ◇ (y₁, y₂) = (3·x₁ + 29·y₁ mod 31, 3·x₂ + 29·y₂ mod 31)
— a single F_31²-linear quasigroup. Medial (since direct products of medial magmas are medial).
Sibling magmas at this size (all four constant-α direct products are now in the DB):
• magma#ba90950a — constant α = 27
• magma#2b8e3c08 — constant α = 15
• magma#5f44e78d — constant α = 23
• magma#fc322ba7 — constant α = 29 (this one)
The fourth b-reinke size-961 entry magma#3c1d2230 has a different template — it is NOT an AG(2, 31) line magma (its 2-generated sub-magmas already span all 961 points), but instead a 16-coset piecewise-linear construction on F_961 (L_0 has cycle structure (1, 60¹⁶), corresponding to 16 cosets of an order-60 subgroup of F_961*).
Display reorder coordinates each point as (i, j) where i indexes one chosen parallel class (C_h) and j indexes another (C_v): each point is the unique intersection of the i-th line of C_h with the j-th line of C_v. Lines within each class are ordered by minimum canonical label. The rendered Cayley table shows a clean 31 × 31 block grid that directly exposes the AG(2, 31) line geometry; it is not (Z/31)²-translation-invariant in the strict sense because the within-class indexing isn't F_31-additive, but the block structure remains the dominant visual feature.
[text written by Claude]
dwrensha · 2026-05-14 23:41:03
dwrensha · 2026-05-14 21:23:46