This principle asserts that for $S$ to effectively precede without conflict, its feerate must equal or surpass that of every individual transaction set $p_i$ within $P$. Additionally, it is crucial that any intersection between $S$ and $p_i$ does not exhibit a higher feerate than $S$, ensuring no discrepancies in transaction prioritization.

The Double LIMO scenario elaborates on this by considering all prefixes of chunks from a linearization $L$, alongside subsets $S_1$ and $S_2$, thereby framing a more complex structure for analysis. Within this context, the LIMO theorem presents itself as a pivotal analytical tool. It stipulates conditions under which a subset $S$ of transactions can be seamlessly integrated as a prefix in a broader linearization $L$ of the graph $G$, without compromising on the fee and size metrics established by $P$. The theorem outlines two critical criteria: first, the feerate of $S$ must not be inferior to that of any $p_i$, and second, intersections between $S$ and any $p_i$ must either have a feerate that is less than or equal to that of $S$ or be non-existent. Fulfilling these conditions guarantees the existence of a linearization where $S$ aligns with the predefined parameters, notably where the diagonal measurement of $L$ concerning the size of each $p_i$ satisfies or exceeds its associated fee requirement.

This theorem not only simplifies the process of identifying such an optimal subset $S$ but also hints at the possibility of discovering the most efficient subset through examining the highest-feerate combinations among all intersections of the $p_i$ sets. This approach offers a methodical way to navigate the complexities of transaction ordering within a graph, ensuring that specific performance benchmarks are met without sacrificing the integrity of the transaction sequence.