This new proposal offers a refined framework for conceptualizing linearizations, including a significant evolution from traditional linearizations to what are now termed partial linearizations. The essence of this advancement lies in viewing linearizations as a collection of topological subsets, each being a subset or superset of another, ultimately encompassing the empty set and the graph itself. This perspective paves the way for a more nuanced interpretation and application of linearizations in transaction graph analysis.

A key innovation within this framework is the simplification of the gathering theorem into a form that is easier to comprehend and utilize. This simplified variant underpins the composition theorem, which elucidates how a set of topologically valid subsets can be merged to form a linearization that inherits the strengths of its components. This theory provides a foundation for understanding how merging and LIMO (Linearization and Merging) processes can be viewed as specific instances of the broader concept of composition within this refined theoretical landscape.

Moreover, the discourse elaborates on several critical theorems and algorithms that bolster the utility of this new linearization framework. Among these, the Full Specialization Theorem asserts that every linearization possesses a fully specialized counterpart, facilitating a method to enhance linearization granularity. The Preordering of Linearizations introduces a system to compare the efficacy of different linearizations using a real function that evaluates their performance based on transaction fees and sizes. Furthermore, the Stripping Theorem and Simple Gathering Theorem offer insights into improving linearizations by manipulating their constituent subsets, highlighting the inherent flexibility and adaptability of this approach.

The Supreme Subset Theorem and Composition Algorithm stand out by providing a methodological backbone for optimizing linearizations. These elements collectively enable a systematic approach to constructing superior linearizations by strategically combining subsets of a graph. This process not only improves the efficiency of linearization but also extends its applicability, demonstrating the robustness and versatility of the proposed framework.

In conclusion, this discussion marks a significant leap forward in the realm of transaction graph analysis through the introduction of a sophisticated linearization framework. By redefining linearizations to include partial linearizations and proposing a series of theorems and algorithms, this approach enhances our ability to manipulate and understand the complex dependencies within transaction graphs. This advancement holds promise for more effective and efficient analysis, optimization, and application in areas where understanding the structure and dynamics of transaction graphs is crucial.