In accordance with the general systems theory of Yu.A. Urmantsev, systems of objects are constructed on the set of prime numbers. The relation between objects is taken as the difference of primes. Each prime number (except the first and the last) is assigned two normalized intervals: the left interval — the difference with the previous prime divided by 2, and the right interval — the difference with the next prime also divided by 2. The third coordinate (sequential number) records the multiplicity of a given combination of intervals. A three dimensional array is obtained, whose projection onto the plane (x, y) reveals a strict periodicity: in the table of residues modulo 6, cells with equal non zero residues (x mod 3, y mod 3) are empty, while the remaining cells contain only 1 or 5. A lemma is proved that explains these regularities. It is shown that the systematisation does not provide a deterministic algorithm for finding the next prime, yet it reveals deep structural constraints.