assignments $b$ such that $0k_0 + 1k_1 +2k_2 = 0 \mod 3, $ and $k_i$ is the number of slots of $x_1+ \dots  +x_k$ filled with a variable $x$ with $b(x)=a(x)+i.$ Thus there are $\sum_{k_1=k_2 \mod 3} {k \choose k- k_1-k_2, k_1, k_2}$ different ways in which an equation can become true under $a, b.$ The following generating function allows us to deal with these possibilities analytically. With $\bf  w_1 \,= \, \exp(2\pi \i/3)$ the primitive third root of unity and $ \bf  w_2\,=w_1^2$ we define $r(x_0, x_1, x_2) = \frac{1}{3} \left[ (x_0\,+\,x_1\,+\,x_2 )^k \,+  \, 
(x_0 \, + \,{ \bf w_1} x_1 + {\bf  w_2} x_2)^k \,+ \, (x_0 \,+ \,{ \bf w_2} x_1 \,+ \, 
{\bf w_1}  x_2)^k \right]$\
then Coeff$[x_1^{k_1}x_2^{k_2}, r(1,\, x_1, \,  x_2)]= {k \choose k-k_1-k_2, k_1, k_2}$ if $k_1=k_2\mod 3$ and $0$ otherwise (easy from properties $ \bf  w_j.)$ In the $\mod 2-$case we use $1/2\left[(1+x)^k \,\, + \,\, (1-x)^k\right]$ instead [@MITZ], Appendix C.\
With the motivation to get an exact threshold of unsatisfiability for a type of constraint whose worst-case complexity is NP-complete, Connamacher/Molloy [@COMO] see also the very recent [@CO] introduce uniquely extendible constraints. A $k-$ary uniquely extendible constraint is a function from 