Universal URM Programs

Theorem
For each integer $$k \ge 1$$, there exists a URM program $$P_k$$ such that:

For each URM program $$P$$ there exists a natural number $$e$$ such that:

For all $$\left({n_1, n_2, \ldots, n_k}\right) \in \N^k$$, the computation using the program $$P_k$$ with input $$\left({e, n_1, n_2, \ldots, n_k}\right)$$

has the same output as the computation using the program $$P$$ with input $$\left({n_1, n_2, \ldots, n_k}\right)$$.

This function $$P_k$$ is a universal program for URM computations with $$k$$ inputs.

Proof
This follows directly from:
 * Kleene's Normal Form Theorem‎;
 * Universal URM Computable Functions‎.