A monogenic fiber product of two different order-49 extensions of the same order-7 quotient. Use coordinates (i,s,t) in F7^3, packed as 49i+7s+t. For representatives 0<=i,j<=6 put kappa(i,j)=floor((18i+8j)/7) mod 7. The operation is
(i,s,t)*(j,u,v)=(4i+j, 4s+u+[i=j=0], 4t+v+kappa(i,j)),
with all three output coordinates reduced modulo 7.
The projections (i,s,t)->(i,s) and (i,s,t)->i+7t identify this as the fiber product of:
* (i,s)*(j,u)=(4i+j,4s+u+[i=j=0]), isomorphic to 0d104541999f3c03dc6265618ea99bf8b14ed79e9753b076747dd2057b6d1584;
* x*y=18x+8y on Z/49, entry bd9c4bf04842e2985c41b595e9243858f92359259c97b118f4c0aa0eb8ec7ca0.
Both have quotient 4i+j on F7. The two projections together are injective, so their Eq677 identities prove Eq677 here.
Independent verification also checked all 117649 Eq677 instances, all 343 Eq255 instances, both projections on every product, and a full generation derivation from element 49. The unique idempotent is 35. The cubic map b(x)=(x*x)*x has image sizes 343,7,1. The magma is right-cancellative, so its fiber matrix is all ones, normal of rank 1. It is nonmedial: (0*0)*(49*0)=230 whereas (0*49)*(0*0)=209.
GAP computes left multiplication group order 7^10, nilpotency class 7, and center order 49. There are 49 left translations of order 7 and 294 of order 49. Thus it differs from the earlier depth-three order-343 example, whose group has order 7^57 and whose rows all have order 7.
This construction represents the two independent offset directions in a rigidity classification: if a finite Eq677 magma has the form (i,s)*(j,t)=(4i+j,f_ij(s)+t) over F7 with any finite abelian fiber A and arbitrary f_ij, then A is an F7-vector space and every f_ij(s)=4s+c_ij. The 49 offset equations have rank 41; changes of fiber origins have rank 6, leaving two classes represented by [i=j=0] and kappa. Thus every such model has offsets U[i=j=0]+V kappa, after changing origins. Independent U,V give this example; complementary fiber directions split off as affine F7 factors. Every one-generated submagma in this entire family has order at most 343, attained here. These models all satisfy Eq255; this does not resolve the unrestricted finite implication.
dwrensha · 2026-09-06 03:49:06