In the companion paper ℕ is the set of positive integers, so r ≥ 1. Its 1-types over B are the atomic types over B, $Partitions.atomicTypes$, and its flips and flip-metric balls are those of $Lax185574.Flips$, with distances measured by shortest paths. A sequence B₀, …, B_m is the list [B₀, …, B_m], in the order of the companion paper, which is the reverse of the order of chains in $Lax185574.Chains$. The bound m ≤ O(r/ε) on the number of steps becomes a bound K · r/ε + K on the length m + 1 of the list.

The algorithm $A$ is chosen before the class, and the sequence is a function $chain$ of r, ε and the graph that $A$ computes, in the sense of $Lax185574.Algorithms$. The companion paper notes that the algorithm of its Damping Theorem does not depend on the class, and the algorithm of this theorem iterates it. The bound on the length in its proof uses only the Damping Theorem for the graph at hand, which holds for every graph with that graph's own constants (every graph lies in the monadically dependent class {G}). The proof gives m ≤ 1 + 2r/ε on every graph with at least two vertices, and graphs with at most one vertex are handled directly. So the length bound is stated for every graph. The input is the triple (r, ε, G) and the output is the list of sets.

The time bound Õ_{𝒞,r}(n³) hides a subpolynomial factor that depends on 𝒞 and r. It is spelled out as O(n^{3+δ}) for every real δ > 0, with a constant c chosen after 𝒞, r, ε and δ. In the companion paper the time bound does not depend on ε, so this is weaker. The width bound is c · (n+1)^ε with the same constant.

The algorithm of the companion paper uses integers of n^{o(1)} bits. Stored in several words, they cost a subpolynomial factor, which the bound O(n^{3+δ}) absorbs, so the word lengths are those of $Lax185574.Algorithms$, with the word-length exponent of the algorithm, which is chosen before the class.