Equation 677 Database

Size 961 = 31² admits several distinct families of Eq 677 magmas. 13 entries are currently in the DB; all satisfy Eq 255. All known constructions use one of two carriers: • F_961 = GF(31²) (additive group (Z/31)²) — for the line magmas and the near-field-style magma • A direct product G × M with G abelian — for the Tao G × M fiber bundles Family A: AG(2, 31) constant-α line magmas (5 magmas, right-cancellative, idempotent). 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. Each line is the F_31 linear magma with parameter α, the same α for ALL 32 parallel classes. Equivalently, the magma is the medial direct product F_31(α) × F_31(α). The 4 valid α-values are the primitive 10th roots of unity mod 31 = roots of Φ_10(x) = x⁴ − x³ + x² − x + 1, namely {15, 23, 27, 29}; all four are present: • α = 15: magma#2b8e3c08 • α = 23: magma#5f44e78d • α = 27: magma#ba90950a • α = 29: magma#fc322ba7 Family B: 16-coset Tao-Type-II piecewise on F_961 (1 magma, right-cancellative, idempotent). magma#3c1d2230. Carrier F_961 = GF(31²) additively; the operation x ◇ y = x + f(y − x) has f with cycle structure (1, 60¹⁶) — 16 cosets of an order-60 subgroup of F_961* (which has order 960 = 16 × 60). f is axis-additive over (Z/31)² but NOT globally F_31-linear (a fingerprint of near-field multiplication). Since 31 is not in the Zassenhaus exceptional near-field list {5, 7, 11, 23, 29, 59}, this is presumably a Dickson near-field construction with the order-2 Frobenius. The 2-generated sub-magmas already span all 961 points (no AG(2, 31) line sub-structure here). Family C: Tao G × M fiber-bundle constructions (8 magmas, non-right-cancellative, idempotent). All 8 are dwrensha submissions of the form Tao(F_31, β=2) × F_31(ζ, ω) for various (ζ, ω) parameters (ζ ∈ {2, 4, 8, 16}, ω ∈ {5, 25}, giving 8 combinations). An abelian base group acts on a fiber via a 2-cocycle; the resulting magma is non-right-cancellative because the fiber action allows column collapses while preserving left bijectivity. Examples: magma#3259f7fb, magma#ecd0e67d, magma#bbe4461e, magma#1335ae05, magma#85c8462b, magma#7ff29f81, magma#67e60649, magma#896897f8. The five Family-A magmas exhaust the constant-α line magmas at size 961 (one per valid α). Non-constant-α AG(2, 31) line magmas (where different parallel classes use different α-values, analogous to many size-121 entries) are NOT yet represented at size 961; by Burnside, the orbit count of α-colorings of 32 parallel classes under PGL(2, 31) would be in the tens of thousands. Display reorders for the four b-reinke entries have been set: Family-A magmas use a 2D AG-coordinate layout (rows = lines of one parallel class C_h, columns = lines of another class C_v), showing clean 31 × 31 block structure; the Family-B magma uses an orbit-of-0 layout under two commuting order-31 fix-free automorphisms (the hidden (Z/31)² additive translations), giving a fully translation-invariant Cayley table. [text written by Claude]