The concept of set-linearizations is introduced, defining a way to organize non-overlapping sets of transactions with topological prefixes. Unlike traditional linearizations, set-linearizations do not demand monotonically decreasing feerate and offer a new perspective on transaction ordering without implicit chunking. The significance of set-linearizations is further elaborated through the introduction of set-feerate diagrams, which depict the cumulative size and fee points of set-linearization prefixes. These diagrams, unlike those for normal linearizations, do not necessarily form a concave function. A critical operation called chunksets is defined for transforming set-linearizations into forms that adhere more closely to desired properties, such as minimal concave functions through prefix points.

Central to the discussion are several theorems that underscore the utility and theoretical underpinnings of set-linearizations and their transformations. For instance, it's proven that applying chunksets to any set-linearization yields a version that is at least as good as the original in terms of diagram comparison. Compatibility between set-linearizations and standard linearizations is also addressed, establishing that chunksets transformed set-linearizations maintain compatibility and do not exhibit superior diagrams compared to their linear counterparts post-chunking. This groundwork facilitates the understanding of how transactions can be optimally organized.

Slope algebra is introduced to provide a mathematical framework for comparing transaction sets based on size and fee. Through definitions like sfpair and rules for slope comparison, the methodology offers a generalized approach to assess transaction feerate efficiency. This algebraic toolset supports intricate analysis and optimization strategies within the Double LIMO algorithm.

The set gathering theorem presents a variation on the gathering theorem, tailored for set-linearizations. It guarantees that moving a high-feerate subset of transactions to the front and then chunking does not degrade the set-linearization's efficiency diagram. This theorem, alongside its corollaries, lays the foundation for the iterative process of improving transaction organization within the Double LIMO framework.

Double LIMO itself is an iterative algorithm designed to optimize transaction linearization by selectively moving subsets of transactions to the forefront based on their feerate. By employing a dynamic selection process involving comparisons of feerate among different subsets and the existing linearization chunks, Double LIMO ensures progressive improvement towards an optimal linearization. The algorithm intricately combines previously defined concepts, such as set-linearizations and slope algebra, to evaluate and reorganize transactions effectively. Each iteration of Double LIMO is guided by theoretical assurances that the chosen subset of transactions will lead to a linearization at least as efficient as the current one.

Furthermore, variations of the Double LIMO algorithm are discussed, suggesting flexibility in its implementation. These variations allow for additional combinations and intersections of transaction subsets to be considered, potentially enhancing the algorithm's ability to find more efficient linearizations dynamically. This adaptability is crucial for managing complex transaction sets and achieving optimal organization.

In conclusion, the development and analysis of Double LIMO, grounded in the newly introduced concepts and mathematical frameworks, represent a significant advancement in transaction organization methodologies. Through rigorous theoretical backing, the algorithm promises to efficiently linearize transactions, ensuring progress toward optimal solutions with each iteration.