The construction sequence is produced by a function $script$ of the graph and the chain, 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)³) and the constant $K$ of the length bound come first, so both constants are absolute. The input is the pair (G, S), with the encodings of $Lax185574.Encodings$. A graph has the vertices $Fin n$, and the notions of chains and construction sequences are applied to its $SimpleGraph$.

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 number w_r is $chainBound G S r$, with π_G the neighborhood complexity $VCDimension.neighborhoodComplexity$. Item 2 is $LocalDefinability.IsRadiusDefinable$ with 2ℓ+1 predicates, w_r parameters and rank 7.

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 is what $Lax185574.DefinableSequences$ asks of its output.