An illustrative example provided shows that splitting a chunk labeled BACFE into [BACE,F] based on the feerate of childless descendants (E and F, with F having a feerate of 11 and the entire chunk's feerate being 13.2) results in an improved structure. This is because, ideally, the feerate of childless-descendants within an optimal chunk should be higher than that of the chunk itself. If a subset of transactions, including all its descendants (referred to as a "bottom" subset), does not possess a higher feerate than the chunk it belongs to, removing this subset could potentially increase the feerate without affecting the overall topology.

The discussion further introduces the concept of applying post-processing steps, such as examining bottom subsets of one or two transactions to assess if their feerate is lower or equal to the chunk’s feerate and then splitting them accordingly. This method could also apply to top transactions with a feerate higher or equal to the overall chunk, aiming to refine chunking methods. The underlying goal is to develop linearization algorithms that inherently avoid inefficient chunking, thereby streamlining the process from the outset.

Moreover, the potential for relaying information about childless descendants to peers as a means of sharing knowledge on linearizations is explored. This approach could be compatible with Erlay, facilitating efficient data transmission through set semantics. This strategy suggests that understanding the composition of childless descendants within each chunk could enable peers to reconstruct similar linearizations, potentially improving the network’s overall efficiency.

Lastly, the text touches upon the idea of merging distinct linearizations. While a specific linearization might appear superior to another, combining two different linearizations could yield even better results. The concept of prefix-intersection merging is mentioned as a technique to integrate distinct linearizations, especially when they share some common transactions. This iterative approach is highlighted as a promising strategy for optimizing transaction processing, indicating ongoing exploration and testing in this area to refine and improve linearization techniques further.