Equation 677 Database

Coordinates Z_7 x Z_7 (display order = 7B+i for element (B,i)). Complete closed form: (B,i) * (C,j) = (4B+C, a·i+b·j) mod 7, where (a,b) depends only on the block pair: (4,1) if B=C=0; (5,4) if B=0, C!=0; (3,1) if B!=0, C=0; otherwise (a,b)=g(C·B^{-1}) with g(1)=g(5)=(3,5), g(2)=(5,3), g(3)=(3,2), g(4)=g(6)=(6,2). So this is a per-pair-varying affine bundle (zero translation part) over the NON-idempotent base F_7(4,1): the base coordinate B is a homomorphism onto F_7(4,1), the fiber over 0 is an F_7(4,1) sub-magma ({42..48}, display row/col 0-6), and the section i=0 ({48,36..41}) is a second F_7(4,1) sub-magma meeting it at the unique idempotent 48. The earlier description (two F_7(4,1) sub-magmas sharing an idempotent + 6 fibers of 6) was this same structure seen without the bundle coordinates: the '6 fibers of 6' are the fibers minus their section points. Not medial, hence not module-linear. Satisfies 255 despite the non-idempotent base. In the display order the table is 49 monochrome 7x7 tiles (block pattern = F_7(4,1)), each tile a linear gradient. [text written by Claude]