Universal URM Computable Functions

Theorem
For each integer $$k \ge 1$$, there exists a URM computable function:
 * $$\Phi_k: \N^{k+1} \to \N$$

such that for each URM computable function $$f: \N^k \to \N$$ there exists a natural number $$e$$ such that:
 * $$\forall \left({n_1, n_2, \ldots, n_k}\right) \in \N^k: f \left({n_1, n_2, \ldots, n_k}\right) \approx \Phi_k \left({e, n_1, n_2, \ldots, n_k}\right)$$.

This function $$\Phi_k$$ is universal for URM computable functions of $$k$$ variables.

Proof
Let $$\Phi_k: \N^{k+1} \to \N$$ be given by:
 * $$\Phi_k \left({e, n_1, n_2, \ldots, n_k}\right) = U \left({\mu z \ T_k \left({e, n_1, n_2, \ldots, n_k, z}\right)}\right)$$

where $$T_k$$ and $$U$$ are as in Kleene's Normal Form Theorem.

Thus we have reinterpreted Kleene's Normal Form Theorem as being about URM computable functions.

This is legitimate, as a URM Computable Function is Recursive and vice versa.

Comment
Thus we can obtain all URM computable functions of $$k$$ variables by letting the value of the first variable of $$\Phi_k$$ to range through all the natural numbers.

So we can think of a corresponding URM program $$P$$ which computes $$\Phi_k$$ as being a universal machine which computes all URM computable functions of $$k$$ variables.

So we now have a recipe for constructing a suitable URM program $$P$$ for this universal machine.

This recipe will be fairly complicated.