Boundary Growth and Relational Decay in a Set-Theoretic Boundary Framework

This publication comprises the manuscript "Boundary Growth and Relational Decay in a Set-Theoretic Boundary Framework", its formal supplementary material, and associated Python implementations. Together, these components define a single, self-contained scholarly work that develops a deterministic, set-theoretic framework in which macroscopic structure, abstraction, and temporal ordering arise from irreversible transformations localized at a boundary. Microstates are generated by a deterministic operator and propagate unchanged through a bulk domain. All structural change occurs at the boundary, where incoming relations are irreversibly aggregated into macrostates. Following aggregation, macrostates undergo a post-aggregative resolution process that monotonically reduces internal relational distinctions and ultimately leads to decay. Irreversibility and directionality are formalized structurally through non-invertible boundary operations, without invoking entropy, probabilistic dynamics, stochasticity, or externally imposed time parameters. The manuscript and supplement fully specify the framework at a formal level, including the set-theoretic operators, ordering relations, and monotonicity properties governing aggregation and resolution. Set theory is used here as a precise representational language for relational structure and transformation. Specifically, the framework uses set-theoretic intervals defined from a set-theoretic alphabet to model the distances between relational components and their evolution over time. These intervals are conceptualized through set-theoretic methods that define how relational distances and structural changes can be measured and tracked within the framework, providing a formal structure for understanding relational decay and boundary-local operations. In this framework, the term "holographic" refers to the conceptual analogy of boundary-local encoding of information, not to imply any physical holography or bulk-boundary duality from string theory or quantum gravity. Additionally, boundary growth refers to the accumulation of relational complexity at the boundary, where incoming microstates are aggregated, increasing the boundary's capacity to store relational information about the system. The framework employs a set-theoretic approach that draws analogies to intervals defined from an alphabet, where relational components are represented as sets, and their distances can be defined through interval relations. This formalism helps describe how relational structures evolve through set-theoretic operations while maintaining a rigorous mathematical foundation for the evolution and decay of macrostates. The associated Python implementations provide explicit constructive illustrations of the formal definitions and transformation rules. They are included for transparency and internal consistency, and to make the abstract constructions inspectable. They do not constitute empirical simulations, numerical experiments, or predictive models, and they are not required for understanding, validating, or applying the formal results presented in the manuscript. No external datasets were generated or analyzed, as the work is entirely theoretical and does not rely on empirical input. This framework is developed further in Alysis, introduced in "Structural Irreversibility and Alysis: A Diagnostic Complement to Entropy in Macrostate Physics" (DOI: 10.5281/zenodo.18165132). Alysis is introduced as a diagnostic tool for evaluating macroscopic irreversibility and assessing whether macrostates remain structurally admissible after irreversible relational decay. The framework presented here serves as the theoretical foundation for Alysis, and this work builds upon these conceptual foundations to introduce a new diagnostic approach to irreversibility in macrostates.