The named polynomial and its exact scope

Define

To make the resource claim nonvacuous, define a partial family ${\mathcal{C}}_{\ell }$ as follows:

• its allocated width is

  $$q_{\ell }=\ell +n(\ell {)}^2+3;$$

• if $\ell =4$, its gate list is the expansion defining $U_4$;

• otherwise, its gate list is empty.

At other lengths this family is not asserted to factor anything.

For every $\ell \ge 0$,

If $\ell \le 128$, subtracting $q_{\ell }$ from $P(\ell )$ gives ${\ell }^2+255\ell +509>0$. If $\ell \ge 128$, the difference is $255\ell +16893>0$. Its gate count is either zero or $175$, also at most $P(\ell )$.

The partial family is length-uniform: compute $n(\ell )$ and its width, and print the one fixed gate list precisely when $\ell =4$. No input integer or factorization is consulted.

This does not evade the all-input issue by changing quantifiers: the partial family explicitly fails the general factoring endpoint at other lengths.