Simple non-RC Eq677 magma, size 265 = 11*24+1, with NON-NORMAL fiber matrix (F F^T != F^T F) of full rank 265. Recipe (see magma#1c1045d74cc5a6cd35415a2d56a5971c for the k=8 prototype): carrier {e} + F_11 x Z/k, base law i*j = 6i + 6j mod 11; off-diagonal fibers (i != j): (i,s)*(j,t) = (i*j, t) if j-i is a nonzero square mod 11, else (i*j, -s-t mod k). Each diagonal fiber together with e is a copy of a seed 677 magma of size k+1 with e as an idempotent, all copies glued along e. For x,y in distinct fibers all four products of Eq677 stay off-diagonal, so the seeds can be chosen independently per fiber. Here k=24 and every diagonal fiber uses the non-right-cancellative size-25 seed magma#d732efd172ca4d7b9967999fff8c2c19229e25d23884fce7dbc53f3cb456aad4 (e = any of its idempotents; Aut is transitive). Compare magma#dc2f657e34e8de11126c640cda74edb6 (size 275, one non-RC diagonal fiber, non-normal but not simple) and magma#d732efd172ca4d7b9967999fff8c2c19229e25d23884fce7dbc53f3cb456aad4 itself (normal). This is the first known Eq677 magma that is simultaneously simple, non-RC, not translation-invariant and non-normal. Verified locally: Eq677 on all instances, simple (each pair (0,b) generates the full congruence), non-RC, not translation-invariant (e is the unique right identity). Constructed by Claude (Fable 5.1) with dwrensha, 2026-09-04.
dwrensha · 2026-09-04 23:15:05