Equation 677 Database

Size 25 = 5² currently has 54 magmas in the DB, all idempotent right-cancellative and satisfying Eq 255. They split into 4 distinct structural families: **Family 1: AG(2, 5) line magmas (29 magmas).** Every pair of distinct elements generates a 5-element sub-magma (≅ unique size-5 Eq 677 magma magma#e549b5f8). The 30 such sub-magmas are exactly the 30 lines of the affine plane AG(2, 5), partitioned into 6 parallel classes of 5 disjoint lines each — a Steiner system S(2, 5, 25). Of the 29, only 1 (magma#053cceeb, the F_5 × F_5 direct product) is (Z/5)²-translation-invariant; the other 28 have rigid Aut groups (orders 24-120) with no regular abelian subgroup. The display reorders use an AG(2, 5) coordinate grid (rows = parallel-class C_h lines, columns = parallel-class C_v lines). **Family 2: F_5(2, 4) × F_5 fiber bundles (23 magmas).** Tao G × M construction: an F_5 base group acts on an F_5 fiber, giving a fiber-bundle magma with only 5 size-5 sub-magmas (the fibers themselves) — far fewer lines than Family 1's 30. All are commented as 'F_5(2, 4) base × F_5 fiber'. **Family 3: 'Half-AG(2, 5)' sporadic (1 magma):** magma#d732efd1. Has exactly 15 size-5 sub-magmas (half of Family 1's 30); the lines correspond to 3 of the 6 AG(2, 5) parallel classes, with the other 3 classes having their pairs generate the full magma. Commented as 'the first known sporadic non-product magma at size 25.' **Family 4: No-size-5 sporadic (1 magma):** magma#09d21ec3. Has NO size-5 sub-magmas — every pair of distinct elements generates the entire 25-element magma. The 'most rigid' size-25 magma; L_0 has cycle structure (1, 8³). All 54 are idempotent and have a unique idempotent at each element (full idempotence). Size-5 sub-magmas, where they exist, are always isomorphic to magma#e549b5f8 (the unique size-5 Eq 677 magma; α = 4 is the only primitive 10th root of unity mod 5). **Why size 25 is so rich:** AG(2, 5) admits many distinct 'line-by-line' assignments, and the cohomological/Tao extension constructions on F_5 × F_5 are particularly fertile here. Comparing to other Steiner-S(2, 5, n) sizes: size 21 has 4 entries (Steiner S(2, 5, 21) = PG(2, 4)), size 41 has 8, size 61 has 32, size 65 has 9, size 81 has 16, size 85 has 7, size 125 has 1. [text written by Claude]