Equation 677 Database

Size-25 = 5² idempotent right-cancellative magma satisfying Eq 677 and Eq 255, in the AG(2, 5) line family. The 30 size-5 sub-magmas of this magma are exactly the lines of the affine plane AG(2, 5); every pair of distinct points lies on a unique line (so this realizes the Steiner system S(2, 5, 25)). Each line is isomorphic to magma#e549b5f8 (the unique size-5 Eq 677 magma over F_5, x ◇ y = 2x + 4y mod 5 — α = 4 is the only primitive 10th root of unity mod 5). The 30 lines decompose into 6 parallel classes of 5 mutually disjoint lines each (covering the 25 elements). Each parallel class is one of the q + 1 = 6 'directions' of AG(2, 5). This member is maximally twisted: no parallel class is a congruence (so there is no quotient onto a size-5 magma), only 158 of the 750 left-translation images of lines are again lines (barely above the 150 forced by closure: each point's translation fixes the 6 lines through it), and the automorphism group is trivial (verified by two independent exhaustive searches) even though all 25 elements have identical row/column cycle-type signatures. Display reorder: the five lines of one parallel class laid out as five contiguous blocks of 5. Each line is a sub-magma, so the five 5×5 diagonal blocks take values only inside their own block (5 consecutive hues in the rendering), and idempotence makes the main diagonal a perfect color gradient. Block order and within-line orders were optimized by simulated annealing on total color variation of the rendered image (canonical order 6818 → 5550). Unconstrained annealing over all orderings independently converges to this exact permutation, so the line-blocked layout is the genuinely smoothest presentation of this table; for calibration, the fully linear family member renders at cost 3500 under the same metric — the remaining off-diagonal roughness is forced by the twist, not by the ordering. There are 29 size-25 magmas in this AG(2, 5) line family currently in the DB. Of these, only 1 (magma#053cceeb, the F_5 × F_5 direct product with both factors using α=4) is fully (Z/5)²-translation-invariant; the other 28 share the same AG(2, 5) Steiner-system structure but break the additive translation symmetry (this one all the way down to a trivial automorphism group). Other size-25 Eq 677 magma families in the DB (NOT in this family): (a) 23 'F_5(2, 4) base × F_5 fiber' fiber-bundle constructions with only 5 size-5 sub-magmas (different from the 30 here); (b) magma#d732efd1, a 'half-AG(2, 5)' sporadic non-product magma with 15 size-5 sub-magmas; (c) magma#09d21ec3, a sporadic non-product magma with NO size-5 sub-magmas at all. [text written by Claude]