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”, a compact formal supplement, and associated Python implementations. Together, these components define a self-contained scholarly work that develops a set-theoretic boundary framework in which macroscopic structure, abstraction, and temporal ordering are treated as consequences of irreversible transformations localized at a boundary.

In the formal construction, microstates are generated by declared operators and propagate through a bulk domain without undergoing structural transformation there. Structural change is localized at the boundary, where incoming relations are aggregated into macrostates. After aggregation, macrostates undergo a post-aggregative resolution process that reduces internal relational distinctions and can lead to decay. Irreversibility and directionality are formalized through non-invertible boundary operations and monotonic loss of internal distinctions. Entropy, probabilistic dynamics, and physical time parameters are not assumed as primitive components of the formal framework, although related scalar descriptions may be applied externally in later analyses.

The manuscript and supplement specify the framework at a formal level, including the relevant set-theoretic operators, ordering relations, and monotonicity properties governing aggregation, resolution, and decay. Set theory is used as a precise representational language for relational structure and transformation. In particular, intervals defined over a set-theoretic alphabet are used to represent relational distances and to describe how these distances are transformed during boundary-local aggregation and post-aggregative resolution.

The term “holographic” is used in a broad structural sense, referring to boundary-local retention, encoding, and transformation of relational information. It does not imply physical holography, AdS/CFT duality, string theory, or quantum-gravitational bulk-boundary duality. Boundary growth refers to the accumulation and transformation of relational complexity at the boundary as incoming microstates are aggregated into macrostates.

The present version uses a compact formal supplement aligned with the revised set-theoretic boundary framework. Earlier exploratory supplements on emergent geometry, structural unification, and broader HBT/HBF developmental steps are not carried forward in this version, because they exceed the conservative scope of the present manuscript.

The associated Python implementations provide illustrative computational realizations of the formal definitions and transformation rules. They are included for transparency, reproducibility, and internal inspection of the proposed architecture. The scripts should not be understood as empirical simulations, predictive physical models, or numerical validation of a physical theory. Where seeded pseudo-random sampling is used, it serves only to generate reproducible internal configurations under fixed rules.

No external datasets were generated or analyzed. The work is theoretical and formal in scope and does not rely on empirical input.

A later and distinct development of related ideas is Alysis, introduced in “Structural Irreversibility and Alysis: A Diagnostic Complement to Entropy in Macrostate Physics” (DOI: 10.5281/zenodo.18165132). In that later framework, structural irreversibility is treated diagnostically through the assessment of whether macrostates remain structurally admissible after relational decay. The present work should therefore be read as a conceptual and formal precursor to later Alysis developments, not as an empirical application of Alysis itself.