Finite-Horizon Structures I: A Minimal Axiomatic Category of Homogeneous Invariants

This article presents a minimal axiomatic framework for describing structured objects when their organization is accessible only through a finite horizon of evaluation or aggregation. Rather than assuming idealized instantaneous access, the work introduces an abstract structural decomposition into three strictly positive parameters, interpreted as magnitude, coherence, and evaluation horizon. These parameters are treated purely formally, without physical, geometric, or dynamical assumptions, and serve as a minimal organizational profile for finite-horizon structure. The central mathematical result is the identification of a unique scalar quantity that remains invariant under admissible homogeneous rescalings of these parameters. This invariant induces a notion of structural equivalence, directional comparison, and admissible transformation between finite-horizon objects. On this basis, the article defines: objects equipped with finite-horizon structural data, strict structure-preserving morphisms acting by uniform rescaling, and proves that these objects and morphisms form a well-defined category. A clear distinction is drawn between exact structural identity, encoded categorically, and more general finite-horizon transformations that act monotonically on structural parameters without preserving strict proportionality. The latter are treated as admissible relations rather than categorical morphisms. The framework is deliberately pre-dynamical and model-independent. It does not propose equations of motion, metrics, or empirical interpretations. Instead, it isolates the weakest structural conditions under which non-trivial invariants persist when evaluation is constrained to finite horizons rather than idealized points. This article is self-contained and can be read independently. Within the broader Ranesis program, it serves as the mathematical foundation for finite-horizon structural analysis, upon which later geometric, dynamical, and domain-specific developments are constructed.