A smaller nonnormal fiber-matrix example (order 231), constructed by gluing five NORMAL order-55 magmas along a common order-11 submagma. It is idempotent, hence satisfies Eq255; it is not a counterexample to the main implication.
For each of five copies use F11 x F5, with (i,s)*(j,t)=(6j-5i,H_ij(s,t)). On the diagonal H_ii=2s+4t. Off the diagonal H_ij=t when j-i is a quadratic residue, and H_ij=-s-t otherwise. In the fifth copy reverse the QR/NQR choice. All arithmetic is in the indicated fields. Identify the five copies of S={(i,0):i in F11}; their induced operations agree. The carrier now has 11+5*44=231 points.
Cover pairs between the five outside groups with TD(5,44), obtained as the product of TD(5,4) and TD(5,11). Explicitly, encode an outside point by (group g, u in F4, v in F11), with u=0,1,2,3 corresponding to fiber s=u+1. For each a,b in F4 and c,d in F11, take the transversal (g,a+g*b,c+g*d) for g=0,1,2,3, and (4,b,d). In F4 the labels use w^2+w+1=0; the F11 slopes are the integers 0,1,2,3. Put the order-5 law g*h=2g+4h on every transversal. Every pair is covered, overlaps have consistent operations, and each Eq677 calculation stays in its chosen submagma.
Nonnormality also has a short matrix proof. Order points as F11 x W, where W is one common point plus five disjoint groups of four points. Let B[z,j]=1 iff j-z is a nonzero nonsquare in F11. Let E_g be 5I-J embedded on the common point and group g. Put U=E_0+E_1+E_2+E_3 and V=E_4. Then the fiber matrix is F=J+B tensor U+B^T tensor V, so
FF^T-F^TF=(B^2-(B^T)^2) tensor (UV-VU),
which is nonzero. In the submitted construction labeling, common points have labels i; outside (g,i,s), s=1..4, has label 11+44g+4i+s-1. Here (FF^T)[0,15]=189 and (F^TF)[0,15]=185. Canonical labels may differ.
All 53,361 Eq677 instances, Eq255, idempotence, and the numerical witness were independently checked by a scalar verifier. Each of the five order-55 submagmas was separately checked to have normal fiber matrix. Corresponding rows and columns of the new fiber matrix still have identical multisets of entries.
dwrensha · 2026-09-05 00:01:54