Equation 677 Database

Twisted fiber bundle of shape 5 x 7 (intermediate-twist variant). Same skeleton as magma#c689a91b and magma#26eaf655: 35 elements partition into 5 disjoint size-7 sub-magmas via a congruence. Quotient is the unique size-5 Eq 677 magma magma#e549b5f8 (affine F_5 line). Each fiber is isomorphic to magma#7981e2df, the size-7 magma with exactly 1 idempotent; the 5 global idempotents 30..34 are the fiber-idempotents, one per fiber. No size-5 sub-magma anywhere (non-split bundle). 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 at position 6. Twist class is intermediate between magma#c689a91b and magma#26eaf655. Classifying off-diagonal fiber slice ops (residue table p1,p2 -> (x*y) mod 7 for fixed fiber pair (f1,f2)): - 11 distinct slice ops total - 5 diagonal pairs share one op (the common fiber operation) - All 20 off-diagonal slice ops occur in 10 pairs of 2: each pair of base-pairs shares a slice op The pairing of base-pairs: (0,1)~(0,2) (0,3)~(0,4) (1,0)~(2,0) (1,2)~(2,1) (1,3)~(2,4) (1,4)~(2,3) (3,0)~(4,0) (3,1)~(4,2) (3,2)~(4,1) (3,4)~(4,3) This pairing is induced by the involution on fibers (1<->2, 3<->4, fiber 0 fixed), which is a genuine automorphism of M. |Aut(M)| = 2. Aut orbits: 7 singletons (the 7 elements of fiber 0 -- this fiber is pointwise-fixed by the involution) + 14 size-2 orbits (the involution pairs corresponding elements across fibers 1<->2 and across 3<->4). Comparison across the size-35 1+3+3+4+12+12 family seen so far: magma#c689a91b: 4 slice ops, |Aut| = 4 (most symmetric twist) this magma: 11 slice ops, |Aut| = 2 (one involution) magma#26eaf655: 21 slice ops, |Aut| = 1 (fully scrambled twist) [text written by Claude]