The construction is a total function on Multicoloured Clique instances, so the map it induces is defined everywhere and the statements about it need no side condition.

Jobs and machines are numbered rather than tagged, because an instance is something a machine is handed and a word presents its jobs in an order. Machine $\binom{k}{2}$ is the validation machine; the edge selection machine of the colour pair $\{a, b\}$ with $a < b$ is machine $b(b-1)/2 + a$, the position of $\{a,b\}$ in the enumeration of pairs by larger element then smaller. That numbering is a bijection onto $\binom{k}{2}$, so the machine count is exactly the paper's.

Jobs come in three blocks. Vertex job $vk + \ell$ is the job of vertex $v$ for colour $\ell$. Then $k^2 n$ colour combination slots, slot $nk + (ak + b)n + z$ belonging to the ordered colour pair $(a,b)$ and the vertex $z$. Then one slot per oriented edge.

The combination block is indexed by all ordered colour pairs, not only the admissible ones, because a closed-form index is what a machine can compute: recovering $(a, b, z)$ from a slot is two divisions, whereas enumerating only the admissible pairs would require a search. A slot whose pair is inadmissible — $a \ge b$, or $z$ coloured neither $a$ nor $b$ — carries an inert job: weight zero and no eligible machine at all, so no feasible schedule can place it and no schedule's weight can mention it. Inert jobs therefore change neither side of the correctness statement, and there are only $k^2 n$ of them.

The edge block cannot be padded the same way. A slot for every pair of vertices would be $n^2$ jobs, and the word the reduction reads has length $\Theta(n + m)$; a reduction whose running time is linear in its input cannot write more than that, so a graph with few edges would put the bound out of reach. The edges are therefore enumerated, and there is exactly one slot per edge.

Two details are done slightly differently from the paper. The paper gives the edge job of $\{u, v\}$, with $\mathrm{colour}(u) = \ell < \ell' = \mathrm{colour}(v)$, the processing time $(k+2)(\pi(v) - \pi(u)) - \ell + \ell'$; here it is one unit shorter, $(k+2)(\pi(v) - \pi(u)) - \ell + \ell' - 1$. With the paper's formula the edge job would start one unit before the job $j_u^{(\ell')}$ ends and the two would conflict; with the shorter one the five jobs that Lemma 1 places on an edge selection machine — the two colour combination jobs, $j_u^{(\ell')}$, the edge job and $j_v^{(\ell)}$ — occupy consecutive intervals, which is what the paper's Figure 1 shows, and Lemma 1's schedule is feasible. Colours are numbered $0, \dots, k-1$ rather than $1, \dots, k$. With colours from $1$ the colour combination job of the $\pi$-first vertex for the pair $(1, k)$ has processing time $(k+2) \cdot 1 - k - 2 = 0$; from $0$ every processing time is positive. Neither change affects the theorem: the weights, the machines and the argument are the paper's.

Processing times and deadlines are clamped, exactly as in Construction 2: the raw formulas satisfy $0 < p \le d$ on every admissible slot, and the clamp is what discharges the two standing conventions without a hypothesis on slots where they are not defined. On an admissible slot the clamp is inactive.

The order $\pi$ is not a parameter but a definition: vertices are ranked by colour, ties broken by index. The paper takes any order refining the colour order and the argument uses nothing else about it; fixing one keeps the construction a function of its input alone.