The Prime Validation Horizon in a Recursive Prime System

This paper presents a structural approach to identifying, validating, and predicting prime numbers within a recursive system derived from Pascal’s triangle. Building on a previously published theoretical foundation, the present study focuses on the operational logic of the Prime Filter Layer (PFL) and illustrates the role of the Prime Validation Horizon (PVH) as a key boundary for deterministic classification. Within the verified range up to 2⋅10^8, the system allows each 1111-code position to be identified either as a structured composite or as a prime, based solely on the state of the X-matrix. By analyzing the system’s projection logic, it becomes clear that all unoccupied positions below the PVH must be prime, since no future product can reach them. The distinction between internal generation through matrix products and external validation through ascending projection is central to this conclusion. A detailed case study involving one prime pair and one prime gap shows that the system is capable not only of explaining known patterns retrospectively, but also of anticipating them in advance. This work does not attempt to replace classical number theory. Instead, it offers a complementary structural perspective in which prime numbers appear as the natural outcome of exclusion. They emerge precisely where all structured combinations are no longer applicable. The author concludes with the belief that what has been revealed so far may represent only the beginning of a deeper structural order.