  Let $M_{d,n}(q)$ denote the number of monic irreducible polynomials in
$\mathbb{F}_q[x_1, x_2, \ldots , x_n]$ of degree $d$. We show that for a fixed
degree $d$, the sequence $M_{d,n}(q)$ converges $q$-adically to an explicitly
determined rational function $M_{d,\infty}(q)$. Furthermore we show that the
limit $M_{d,\infty}(q)$ is related to the classic necklace polynomial
$M_{d,1}(q)$ by an involutive functional equation, leading to a phenomenon we
call liminal reciprocity. The limiting first moments of factorization
statistics for squarefree polynomials are expressed in terms of a family of
symmetric group representations as a consequence of liminal reciprocity.
