A NON-IDEMPOTENT Eq677 magma with a NONNORMAL fiber matrix, built using a LINEAR base and AFFINE operations in EVERY fiber block. Order847=121*7. It still satisfies Eq255, so it is not a counterexample to the main implication. Exactly121 elements are idempotent;726 are not.
Carrier F11^2 x F7. Write base vectors i=(i1,i2), j=(j1,j2), and define
(i,s)*(j,t) = (6j-5i, H_ij(s,t)).
Select the two coordinate axes in F11^2. If i,j are distinct and lie on the same selected axis, let delta be j-i in that axis coordinate. Set H_ij(s,t)=t when delta is a nonzero square in F11, and H_ij(s,t)=-s-t otherwise. For every other pair, including i=j, set H_ij(s,t)=4s+3t. Fiber arithmetic is modulo7.
Every fiber block is affine: its (left,right) coefficients are (0,1),(-1,-1), or(4,3). No nonlinear exceptional core is used.
Why Eq677 holds: for distinct base inputs, all four pairs in its evaluation remain on the same affine line and have distinct entries. On a selected axis the fiber trace is 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. On any other line the four operations are C(s,t)=4s+3t, a standard seven-element Eq677 law. Equal base inputs also stay within a C fiber.
Fiber matrix explanation: let K be the121-by121 matrix formed by embedding the Paley tournament matrix B[z,j]=1 iff j-z is a nonzero nonsquare on each of the two axes. The axes share only the origin, and K has no diagonal entries. Put P=7I7-J7. Then
F = J847 + K tensor P,
FF^T-F^TF = 7*(KK^T-K^TK) tensor P.
K is the sum of two individually normal tournament matrices, but is nonnormal because the axes intersect. For the points (0,1) and(1,0), the KK^T entry is1 and the K^TK entry is0. This gives nonnormality directly. K has rank21, hence F has rank127. Corresponding rows and columns of F still have matching entry multisets.
Pack (i1,i2,s) as7*(11*i1+i2)+s. In these construction labels, 1*1=0, and (FF^T)[7,77]=889 whereas(F^TF)[7,77]=847. Canonical server labels can differ.
All717409 Eq677 instances, Eq255, the idempotent count, and both witnesses were independently checked with a separate scalar verifier. The full block-matrix and commutator identities were also checked exactly.
Constructed and verified with OpenAI Codex, 2026-09-04. This shows that affine fiber operations over a linear base do not by themselves force normality of the fiber matrix.
dwrensha · 2026-09-05 02:09:34