The construction sequence is produced by a function $script$ of r, ε and the graph, as a script of $Lax185574.ConstructionSequences$, and the algorithm $A$ computes this function, in the sense of $Lax185574.Algorithms$. The algorithm, the function and the absolute constant $K$ of the bound O(r/ε) on the number of predicates come first, so none of them depends on the class. On every graph, the script gives a construction sequence of the graph. The input of the algorithm is the triple (r, ε, G), with the encodings of $Lax185574.Encodings$.

The constant δ of the paper is $a$. It is chosen after the class and before r and ε. The time bound Ô_{𝒞,r,ε}(n³) is spelled out as $n^(3+η)$ for every real η > 0, with the constant $c$ depending on 𝒞, r, ε and η. The same $c$ bounds the sequence. The bounds on the sequence do not involve η, so a single η, say η = 1, gives a constant for them. The number of unary predicates is at most $⌈K · r/ε⌉ + K$, which is O(r/ε) in the sense of the paper, including its additive constant.

Every graph belongs to some monadically dependent class, for instance the class containing it alone. So the algorithm computes the script of every graph, at every word length admissible for the constant of some class that contains it.

Every number in the encoding of the script is at most the length of the encoding plus n ($Script.HasSmallNumbers$). The paper does not say this. The algorithm can ensure it by giving the vertices of the initial partial graph the colors 0, …, n − 1, since steps only use colors that occur.