Equation 677 Database

Twisted (NOT direct) fiber bundle: F_11(10, 2) base × F_5 fiber. Size 55 = 5 * 11. Fully idempotent. Left-cancellative but NOT right-cancellative -- this is the key indicator that the bundle is non-trivial: a direct product of two RC magmas would itself be RC. Structure: the 55 elements partition into 11 disjoint size-5 sub-magmas (the "fibers"); the fiber partition is a congruence. Quotienting by the fibers gives a size-11 magma = magma#15fbff50 (the linear F_11 magma x ◇ y = 10x + 2y mod 11, which is right-cancellative). Each fiber is isomorphic to magma#e549b5f8 (the F_5 affine line, also RC). But the bundle is TWISTED: a true direct product F_11(10,2) × F_5 (Tao G × M product) would satisfy T((x1, x2), (y1, y2)) = (T_F11(x1, y1), T_F5(x2, y2)), making the whole magma a quasigroup since both factors are. Here, in the suggested reorder (11 consecutive 5-blocks), the diagonal 5×5 blocks correctly show the F_5 fiber operation, and the off-diagonal blocks reflect the base operation -- but the fiber operation embedded in off-diagonal blocks differs from a constant action across (f1, f2) pairs, breaking right-cancellation. In addition to the 11 size-5 fiber sub-magmas, there is exactly 1 size-11 sub-magma in M (a "section" of the bundle, an embedded copy of F_11(10, 2) hitting each fiber in exactly one point). The full sub-magma lattice: 1 size-11, 11 size-5, plus singletons and M itself. The construction is the "Tao Type II fiber bundle" or "twisted G × M" recipe at sizes that are products of two primes admitting Eq 677 linear magmas (5 and 11 both ≡ 1 or 5 mod 20). Compare with the various size-65 (= 5*13) bundles in the DB. [text written by Claude]