The prime-tuple hypothesis for simultaneous primality of linear forms

Let $(a_i)$ and $(b_i)$ be finite collections of integers with $a_i\neq 0$ for every $i$. The linear forms are subject to a congruence obstruction if there is a prime $p$ such that, for every integer $x$, at least one of the values $a_i x+b_i$ is divisible by $p$.

Prime-tuple hypothesis. If no congruence obstruction exists, then there are infinitely many values of $x$ such that every $a_i x+b_i$ is prime.

This hypothesis is generally believed to be true, but is far beyond current methods; it is used in the paper to construct additive and difference decompositions of the signed primes.