Universal partial computable function ($U(e,\mathbf{x})\simeq {\phi }_e(\mathbf{x})$)

A universal partial computable function simulates the program with index $e$ on input $\mathbf{x}$, reproducing both its output and its divergence. Decode the finite program and repeatedly apply its transition rule. This gives a partial computable function of the code and inputs. There can be no total computable evaluator whose rows are precisely all total computable functions, because $n\mapsto U(n,n)+1$ would be a missing diagonal row.