Let $(F_1, \dots, F_m)$ be a family of subsets of $\{0, \dots, n-1\}$ and $k$ a solution size. Its numbers are the list $n, m, k$ followed, for every set $F_j$ in turn, by $|F_j|$ and the elements of $F_j$ in increasing order. The binary word of the instance in lax-496464 writes exactly these numbers, each as its binary digits preceded by their number in unary, so a word RAM program reads the numbers off the word in linear time.

From the numbers a second program writes the standard word of the zoo instance of the construction of lax-496464 (concept $JitFlowShop$) with $k + 1$ machines, together with the threshold $N - R m (2k - 1)$, where $N$ is the number of jobs of the construction and $R m (2k - 1)$ its target. It walks the numbers once, storing the membership pairs $(j, i)$, $i \in F_j$, in two arrays, and then writes the jobs one by one, computing for each its family, segment, set and element by division with remainder, as the construction numbers them.

Theorem. There is a function on lists of numbers computable in polynomial time on the word RAM that maps the numbers of every instance with $k \le n \le \ell + 4$, where $\ell$ is the length of the list, to the standard word of the zoo instance of its construction with the threshold $N - R m (2k - 1)$.

The bound $n \le \ell + 4$ keeps the number of jobs, which exceeds $2Rmn$, polynomial in the length of the list; the instances that the NP-hardness of Hitting Set (lax-496464) produces satisfy it.

For $2 \le k \le n$ the construction has $R m (2k - 1)$ just-in-time jobs exactly when the family has a hitting set of size $k$ (lax-496464, Theorem 1), and then the zoo instance has a schedule with at most $N - Rm(2k-1)$ jobs not just in time. With $Q = (k-1)(n+1)$ and $g \le Rm$, every number of the word is at most $g^2 Q + (2g + 1) Q + n + 1 \le 64 (N + 1)^4$, and the word has more than $3N$ entries. Composing the reduction of every language in NP to Hitting Set with the two programs therefore gives words whose numbers are bounded by a fixed polynomial in their length.

Theorem. $FF(1,m) \mid\mid \sum Z_j$ is strongly NP-hard under the standard reading.