Equation 677 Database

Twisted fiber bundle of shape 5 x 7 (fully twisted variant). Same skeleton as magma#c689a91b: the 35 elements partition into 5 disjoint size-7 sub-magmas via a congruence. Quotient is the size-5 affine F_5 line magma magma#e549b5f8 (unique Eq 677 magma at size 5). Each fiber is isomorphic to magma#7981e2df, the size-7 magma with exactly 1 idempotent. The 5 global idempotents 30..34 are exactly the 5 fiber-idempotents, one per fiber. 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. Diagonal 7x7 blocks reveal the common fiber operation; off-diagonal blocks reveal the bundle's twist. Bundle is non-split: there is no size-5 sub-magma anywhere in M, so the 5 idempotents do not form a sub-magma (T(30,31) = 9, not 31). What distinguishes this from magma#c689a91b is the *twist class*. Classifying off-diagonal fiber slice ops (the residue table p1,p2 -> (x*y) mod 7 for fixed fiber pair): - this magma: 21 distinct slice ops (5 diagonal pairs share one op; all 20 off-diagonal slices are pairwise distinct), so the twist is essentially "fully scrambled". - magma#c689a91b: only 4 distinct slice ops (much more symmetric twist). Reflecting the heavier twist, |Aut(M)| = 1 (trivial — only the identity automorphism), versus |Aut(magma#c689a91b)| = 4. Every element is a singleton Aut-orbit. R_0 cycle signature 1+3+3+4+12+12 and L_0 signature 7+14+14 match magma#c689a91b exactly, so the cycle-pattern fingerprint cannot distinguish the two; the discriminating invariants are the twist class and |Aut|. [text written by Claude]