$D^k$ to true, false with the property: Given values from $D$ for any $k-1$ argument slots there is exactly one value for the remaining slot which makes the constraint true. (The $k> 8$ in the following result can be eliminated at the price of some additional technical effort.)

\[uni\] Let $F(n, p)$ be the random formula of uniquely extendible constraints: Each constraint is a random $k-$tuple of variables and a $k-$ary uniquely extendible constraint over $D$ and we pick with probability $p.$ For $|D|=4 $ and $p < (T- \varepsilon)/n^{k-1}$ $F(n, p)$ is satisfiable whp. for $k>8.$

The threshold $T/n^{k-1}$ is proved for $k=3$ and $|D|=4$, cf. [@CO] remark following Theorem 8. Our proof uses the technique as in the $\mod 3-$case, however the details are different. One of the contributions making is the generating polynomial\
$ p(x)=\frac{1}{d} \left[(1+x)^k+ (d-1)(1-\frac{x}{d-1})^k \right],$ as $r(x_0, x_1, x_2)$ above, not used before.

Motivation
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Many computational problems can be naturally formulated as conjunctions of constraints. And we are interested to find a solution of this conjunction. Algorithmic properties of these conjunctions are considered in theoretical research (with remarkable results e. g. in the realm of approximation[@HA]) and applied research, e. g. [@MEI]. An 