A singly generated, NON-IDEMPOTENT Eq677 magma with a NONNORMAL fiber matrix, of order 469=7*67. It has exactly one idempotent and still satisfies Eq255. In particular, restricting attention to singly generated magmas does not restore normality of the fiber matrix.
Let C be the canonical table of the existing simple order-67 source magma#52769e419ce32f9a2e93e259fd91c2b1173a5fac43fe38951783213649d3de1a. Let p swap its labels 0 and 1 and fix every other label. Set D(s,t)=p(C(p(s),p(t))); thus D is just a relabeled copy of C.
On F_7 x {0,...,66}, define
(i,s)*(j,t) = (4i+3j mod 7, H_ij(s,t)),
H_00 = D, and H_ij = C for every other pair.
Why Eq677 holds: the base F7(4,3) satisfies Eq677. For its instance with y=i and x=j, the four base pairs, inside out, are (i,j), (4i+3j,i), (j,5i+5j), and (i,i+5j). Each is an invertible linear function of (i,j). Therefore all four pairs are (0,0) when i=j=0, and none is (0,0) otherwise. Every fiber instance consequently uses either D four times or C four times. Both satisfy Eq677.
In construction labels (i,s)->67i+s, the element 67=(1,0) generates all 469 elements. This was checked both by iterative set closure and by an independent scalar checker that records a multiplication derivation for each of the other 468 elements. The unique idempotent is 66=(0,66); 0*0=56 witnesses non-idempotence. The square image has 67 elements; the magma is not unipotent.
Matrix explanation: write FC and FD for the fiber matrices of C and D, and E for the 7x7 matrix with E[0,0]=1 and all other entries zero. Then
F = J_7 tensor FC + E tensor (FD-FC).
Although FC and FD are both normal, their relative placement breaks normality of F. An exact witness in construction labels is (FF^T)[0,68]=979 and (F^TF)[0,68]=954. Corresponding rows and columns still have identical multisets of entries.
All 219,961 Eq677 instances, all Eq255 instances, the unique idempotent, generation by element 67, and the integer nonnormality witness were independently checked before submission. Server canonical labels may differ. This is not a counterexample to finite Eq677=>Eq255.
Constructed and verified with OpenAI Codex, 2026-09-05, using the existing order-67 source credited above.
dwrensha · 2026-09-05 21:02:11