Equation 677 Database

Twisted fiber bundle of shape 5 x 7. Structure: the 35 elements partition into 5 disjoint size-7 sub-magmas (the "fibers") via a congruence relation. Quotienting by the fibers yields a size-5 magma magma#e549b5f8, the unique Eq 677 magma at size 5 (the affine F_5 line magma, x*y = x + 2(y-x) mod 5). Each fiber is isomorphic to magma#7981e2df, the non-fully-idempotent size-7 magma (1 idempotent + 6 non-idempotents). The five global idempotents 30..34 are exactly the five fiber-idempotents (one per fiber), so the non-idempotent count is 5 of 35 (matching idem=False). In the suggested reorder, the 35 elements are arranged as five consecutive 7-blocks (positions 0..6, 7..13, 14..20, 21..27, 28..34), one per fiber, with each block's idempotent placed at position 6 (last). The diagonal 7x7 blocks then expose the common fiber operation, and off-diagonal blocks show the bundle's twist. Bundle is non-split: there is no size-5 sub-magma anywhere in M, in particular the 5 idempotents are not closed under the operation (e.g. T(30,31) = 16, not 34). The base operation Q on the 5 fibers permutes them, but the action restricted to the idempotent section fails to be a homomorphism on the nose. Twist structure: classifying off-diagonal "fiber slice ops" (the residue table p1,p2 -> (x*y) mod 7 for fixed fiber pair) gives just 4 distinct slice ops over the 25 base-pairs: - 13 pairs share one op (including the 5 diagonal pairs) - 4 pairs (0,j) for j>0 share another op - 4 pairs (i,0) for i>0 share another - 4 pairs ((1,3),(2,1),(3,4),(4,2)) share the last This non-trivial twisting is why |Aut(M)| = 4 is small (a true direct product F_5 x M_7 would have |Aut(F_5)| * |Aut(M_7)| = 20 * |Aut(M_7)|). Aut orbits: 7 singletons {0,1,2,3,4,5,30} = the special "all-singletons" fiber (the one through 30) + 7 size-4 orbits, one per residue-class of size-4 in each of the other 4 fibers. The 4 non-special fibers each form one size-4-orbit transversal (1 element per Aut orbit). Compare with the size-96 magma magma#b1cfacfa (6 disjoint size-16 petals, non-split) and size-127 / size-181 (single piece, Tao Type II). This size-35 magma is the smallest "5-fiber x 7-fiber" twisted bundle in the DB. [text written by Claude]