A NON-IDEMPOTENT Eq677 magma with a NONNORMAL fiber matrix, of order 539=11*49. Exactly 11 elements are idempotent; the other 528 are not. It still satisfies Eq255, so this is not a counterexample to the main implication.
Carrier F_11 x F_7^2. Pack a fiber vector (u,v) as 7u+v. Define
(i,s)*(j,t) = (6j-5i mod 11, H_ij(s,t)).
For i!=j, take H_ij(s,t)=t if j-i is a nonzero square in F11, and H_ij(s,t)=-s-t otherwise, with vector arithmetic in F7^2. For diagonal i=j!=0 take H_ii(s,t)=4s+3t. At i=j=0, use the canonical table of the non-idempotent, non-right-cancellative order-49 source magma#d50b3565f55696fb14ff77cb5b87148a39c99833e06e009f0d6349acb1e2c81d.
Why Eq677 holds: distinct base inputs make all four products stay off the diagonal. The four fiber operations are A,A,B,B or B,B,A,A, where A(s,t)=t and B(s,t)=-s-t; both chains return the required input over any abelian group. Equal base inputs stay in a single diagonal fiber. Each diagonal law satisfies Eq677 independently; 4s+3t is the product of two standard non-idempotent seven-element models.
Matrix explanation: let D be the fiber matrix of the order-49 source, K=D-J_49, B[z,j]=1 iff j-z is a nonzero nonsquare in F11, and E have its sole nonzero entry E[0,0]=1. Then
F = J_539 + B tensor (49I_49-J_49) + E tensor K,
FF^T-F^TF = 49*((B-B^T)E-E(B-B^T)) tensor K.
D is symmetric, and K is nonzero, so the commutator is nonzero. These two complete matrix identities were also verified using integer arithmetic.
In construction labels (i,s)->49i+s, 0*0=47, witnessing non-idempotence. Nonnormality witness: (FF^T)[0,53]=735, while (F^TF)[0,53]=441. Canonical server labels can differ.
All 290,521 Eq677 instances, Eq255, the 11-element idempotent set, and both witnesses were independently verified by a separate scalar checker before submission. Corresponding rows and columns of F still have identical multisets of entries.
Constructed and verified with OpenAI Codex, 2026-09-04. This adapts the independent-diagonal mechanism of magma#dc2f657e34e8de11126c640cda74edb6 using a non-idempotent core.
dwrensha · 2026-09-05 00:11:16