An initial concern was raised regarding the efficacy of the Double-LIMO workflow. It was identified that starting with an arbitrary linearization and applying Double-LIMO, which incorporates a two-step process involving selecting the best remaining ancestor set ($S_1$) followed by a bounded search ($S_2$), does not necessarily improve or even match the quality of linearization achieved through a pure ancestor set-based approach. This revelation underscores the complexity of achieving optimal linearization and highlights the potential for outcomes that are either incomparable or inferior to those derived from simpler, ancestor set-focused strategies.

Despite this setback, there emerges a silver lining that shifts the perspective on what is essential for effective linearization. The critical insight gained is a recognition that the ultimate goal is not to surpass the theoretical ideal of pure ancestor sort linearization but rather to meet the practical requirements of Child Pays for Parent (CPFP) transactions and the existing network protocols. In this regard, the current version of Bitcoin Core's block builder already ensures a satisfactory level of quality without resorting to chunking methods. Furthermore, it is noted that LIMO, alongside bounded search techniques that incorporate ancestor set-based presplitting, successfully achieves this standard, thereby aligning with the network’s operational needs.

The most promising development comes with the introduction of a new, simplified Double (and Triple) LIMO methodology. Preliminary proofs suggest that this revised approach is effective, meaning it produces results that are at least as good as the input provided. More importantly, it ensures that transactions included at each step of the process are compatible with the objectives defined by the $S_1$ and $S_2$ parameters identified within the same step. This advancement signals a significant step forward in refining transaction linearization techniques, promising enhanced efficiency and reliability in meeting the network's transaction processing requirements. Further elaboration on this proof and its implications is anticipated, indicating a continued evolution in the understanding and optimization of transaction linearization methodologies.