The construction sequence is produced by a function $script$ of the graph and the merge sequence, as a script of $Lax185574.ConstructionSequences$, and the algorithm $A$ computes this function, in the sense of $Lax185574.Algorithms$. The algorithm, the constant $c$ of the time bound O((ℓ+1) · (n+1)²) and the constant $K$ of the length bound come first, so both constants are absolute. The input is the pair (G, merge sequence). A graph has the vertices $Fin n$, and the notions of merge sequences and construction sequences are applied to its $SimpleGraph$.

A merge sequence is one in the sense of $Lax185574.MergeSequences$, so 𝒫₁ is the partition into singletons and 𝒫_ℓ = {V(G)} by its definition.

The width of item 1 is the radius-r width of $Lax185574.ConstructionWidth$, which counts parts. The length bound O(n w₁) is spelled out as K · (n+1) · (w₁+1). The radius-r width of the merge sequence is $MergeSequences.mergeWidth$. The definability hypothesis of item 2 is $BetterMergeSequences.StagesDefinable$ for some two formulas of quantifier rank at most 5, and the conclusion is $LocalDefinability.IsRadiusDefinable$ with 2ℓ+1 predicates, w_r parameters and rank 7.

Deviation from the paper. Both bounds use max(1, w_r) instead of w_r. On a graph with one vertex the merge sequence of length 1 has width 0, the maximum over an empty range, while every construction sequence has width 1. For n ≥ 2 the two agree, since every merge sequence has length at least 2 and width at least 1.

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. An algorithm can ensure it by giving the vertices of the initial partial graph the colors 0, …, n − 1. The bound lets the output be passed on as in $Lax185574.DefinableSequences$.