Equation 677 Database

A sharp cubic-contraction example with a nilpotent left multiplication group. Coordinates are (a,i,s) in F7^3, packed as 49a+7i+s. Put delta=1 if a=b=0 and 0 otherwise. The operation, with all arithmetic modulo 7, is (a,i,s)*(b,j,t)=(4a+b, 4i+j+4delta, 4s+t+[i+delta=0 and j+delta=0]). Construction: start with C on F7^2, (i,s)*(j,t)=(4i+j,4s+t+[i=j=0]), which is isomorphic to the existing entry 0d104541999f3c03dc6265618ea99bf8b14ed79e9753b076747dd2057b6d1584. Let p(i,s)=(i+1,s), an actual left translation of C, and D(s,t)=p^-1(C(p(s),p(t))). Over the outer base F7 with operation 4a+b, use D at base pair (0,0) and C everywhere else. The four base pairs visited by Eq677 are zero simultaneously, so each fiber trace uses just one of the two isomorphic operations. Independently checked all 117649 Eq677 instances and all 343 Eq255 instances. This is a quasigroup: its fiber matrix is all ones, hence normal of rank 1. Its unique idempotent is 47=(0,6,5); element 54 generates the whole magma. The cubic map b(x)=(x*x)*x has successive image sizes 343,4,2,1, with orbit 54 -> 5 -> 42 -> 47. Thus b^3 is constant but b^2 is not. Every left translation has order 7. The left multiplication group is contained in the triangular permutation group (b,j,t) -> (b+c,j+f(b),t+g(b,j)), of order 7^57. GAP gives equality, nilpotency class 49, and center order 7. More generally, for a finite Eq677 magma with nilpotent left multiplication group and order 7^d, the cubic map becomes constant at a unique idempotent after at most d iterations. Proof: the nontrivial commutator-orbit quotient is an elementary abelian 7-group with operation 4u+v+c; b maps into its idempotent fiber, a proper submagma whose left multiplication group is again nilpotent. Induct on d. Repeating the conjugate-at-(0,0) construction above attains this bound at every d. This does not prove Eq255 in the general nilpotent case: eventual constancy of b does not establish b(x)*x=x.