 Factorial graph formula via remainder coding

The graph of the factorial function has a fixed-length first-order formula in the language of ordered rings interpreted on the natural numbers. A remainder code records $a_0=1$, $a_n=y$, and $a_{i+1}=(i+1)a_i$ for every $i<n$. Uniqueness of remainders proves $y=n!$; sufficiently large pairwise-coprime moduli and the Chinese remainder theorem supply a code for every input. Formula length is independent of the input variable's value, although substituting a numeral adds its encoding length.