A Recursive Architecture for Explaining the Distribution of Prime Numbers

This work presents a mathematical architecture for explaining the distribution of prime numbers. It is based on a binary coding system that operates through integer divisibility conditions derived from Pascal’s triangle. The resulting binary codes identify structurally relevant positions within the set of natural numbers. These positions form what is referred to as the Prime Filter Layer (PFL), which is exclusively occupied by structured numbers and prime numbers. A recursive multiplication system, known as the X-matrix, generates all structured numbers in advance. The remaining unoccupied positions within the PFL necessarily correspond to prime numbers. The entire system is deterministic and reproducible. It is capable of identifying prime numbers and predicting prime gaps based solely on structural logic, without relying on classical divisibility tests. This recursive mechanism allows not only for the explanation of prime gaps and the distribution of natural numbers, but also for the advance identification of prime numbers, precisely at the points where the system no longer permits structural reconstruction. The architecture has been computationally validated up to 2 × 10⁸. It was shown that every prime number within this range meets the structural criteria, while all non-primes can be explained as products of earlier matrix values. These findings confirm the existence of a deterministic structure that not only models, but systematically explains the distribution of prime numbers.