Hyper Comparison Categories: structural foundations for multi-level comparison and height inequalities. This note formulates a categorical framework for organizing comparison problems across multiple observational or coarse–grained levels. A hyper comparison category consists of projection functors, an idempotent saturation operator, and invertible 2-cells measuring discrepancies between saturation and observation. These structures produce quantitative invariants such as defects, local supports, and height-type functionals. Under mild locality and stability assumptions, defects admit local decompositions and satisfy propagation properties along towers of projections. As a consequence, inequalities of the form defect ≤ (1+ε)·height + C follow from purely structural considerations once suitable local bounds are available. The purpose of this work is conceptual and organizational: the framework serves as a reusable structural template for comparison and height inequalities in a broad range of mathematical contexts.
jikim4222-kjn et al. (Thu,) studied this question.