Existential bounded representation of a primitive recursive function

The graph of every primitive recursive $f:{\mathbb{N}}^k\to \mathbb{N}$ has a formulain the language of ordered rings, where every quantifier inside $\delta$ is bounded. Composition uses existentially quantified intermediate values, while primitive recursion uses a boundedly checked code for the finite sequence of intermediate values.

The Gödel beta function codes finite sequences by remainders:Choosing $c$ sufficiently large and divisible by the relevant small integers makes the moduli pairwise coprime, so the Chinese remainder theorem codes any prescribed finite sequence.