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

This article develops a minimal axiomatic framework for describing structured objects whose organization is accessible only through a finite horizon of evaluation, aggregation, or resolution. Rather than assuming idealized instantaneous access, the paper introduces a strictly positive structural triplet, interpreted abstractly as magnitude, coherence, and evaluation horizon. This triplet is treated in a fully model-independent way: no physical, geometric, or dynamical interpretation is presupposed. It serves as the minimal structural profile required to represent finite-horizon organization. The article then identifies a canonical homogeneous scalar quantity associated with this triplet. This quantity is not interpreted as an absolute numerical invariant under simultaneous positive rescaling, but as the simplest scalar representative attached to a projective structural class. The genuinely preserved content therefore lies not in an isolated numerical value, but in the projective equivalence class generated by the structural triplet. On this basis, the paper introduces Y-objects, projective structural equivalence under common positive rescaling, normalized representatives, and the logarithmic differential content naturally associated with projective Y-structure. It also defines weighted comparison relations and strict structure-compatible morphisms, called Y-morphisms. A central mathematical result is that Y-objects and Y-morphisms form a well-defined category, denoted Y-Abs. At the same time, the paper clearly distinguishes exact structural identity from more general admissible structural transformations. The latter act monotonically on the structural parameters but do not preserve strict proportionality, and are therefore treated as structural relations rather than categorical morphisms. The framework is deliberately pre-dynamical and model-independent. It does not introduce equations of motion, geometric background, or empirical interpretation. Its purpose is to isolate the weakest structural conditions under which meaningful organization persists 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 provides the first minimal categorical foundation for finite-horizon structural analysis, intended as a basis for later geometric, differential, dynamical, and domain-specific developments.