The fixed machine coding gives an acceptable numbering

This family is a numbering of the partial computable unary functions. If $f$ is partial computable, some deterministic one-tape Turing machine computes it. Relabel that machine's finite state set and tape alphabet by integers so that the start, accept, and reject states become $0,1,2$ and the symbols become the normalized alphabets of [L3]. The resulting coded machine has some code $e$, and by construction ${\phi }_e=f$.

The numbering is universal. On input $(e,x)$, first run the total decoder from [L2]. If $e$ is malformed, diverge forever. If $e$ decodes to a machine $M_e$, simulate $M_e$ on input $x$ and output exactly what $M_e$ outputs if it halts. This produces a partial computable binary function $U(e,x)$ with $U(e,x)\simeq {\phi }_e(x)$.

For hard-wiring, given $e$ and $a$, first run the total decoder [L2]. If $e$ is malformed, construct a fixed machine $N_{e,a}$ that diverges on every input. Otherwise construct a coded machine $N_{e,a}$ that on input $x$ first computes the pair code $<mi\; mathvariant="normal">⌜</mi>a,x{<mi\; mathvariant="normal">⌝</mi>}_{\mathrm{s}\mathrm{e}\mathrm{q}}$, then simulates the machine coded by $e$ on that code, and copies out the simulated output if the simulation halts. By [L2] and [L3], this case-defined map $(e,a)\mapsto <mi\; mathvariant="normal">⌜</mi>N_{e,a}<mi\; mathvariant="normal">⌝</mi>$ is total and computable.

By construction, for every $x$ one has ${\phi }_{<mi\; mathvariant="normal">⌜</mi>N_{e,a}<mi\; mathvariant="normal">⌝</mi>}(x)\simeq {\phi }_e(<mi\; mathvariant="normal">⌜</mi>a,x{<mi\; mathvariant="normal">⌝</mi>}_{\mathrm{s}\mathrm{e}\mathrm{q}})\mathrm{.}$ So the function $s(e,a):=<mi\; mathvariant="normal">⌜</mi>N_{e,a}<mi\; mathvariant="normal">⌝</mi>$ is the required hard-wiring operation from [L1].

Steps 1.1, 1.2, and 2.1 show that $({\phi }_e)$ is a numbering, is universal, and has the required hard-wiring map. Therefore the fixed machine coding gives an acceptable numbering.