The Modulo-30 Sieve and the Prime Periodicity Model: From Number-Theoretic Structures to Interdisciplinary Applications

Based on the eight residue classes coprime to 30 (referred to as the 8 orbits), this paper develops a statistical sieve method and discovers that during the sieving process the candidate set size follows a logarithmic decay law S(n) = A − B ln n. The method is verified on a quantum simulator using a 20‐qubit circuit (factor- ing the composite number 1891 in 18 seconds). Building on this, a prime pe- riodicity model is proposed, revealing that every prime factor possesses a fixed period (equal to the prime itself) and a phase (residue modulo 30); composite numbers can be viewed as superpositions of such periodic signals. The model is further extended to post‐quantum cryptography (phase hiding), radar identifica- tion (low‐probability‐of‐intercept waveforms), and room‐temperature quantum er- ror correction (periodic measurement replacing energy measurement). The research was inspired by the ancient Chinese philosophical classic Tao Te Ching. The work was carried out independently; after its completion a literature search revealed that Mr. Gary William Croft had independently observed the structure of the eight residue classes modulo 30 as early as 1993. The author hereby expresses respect to this pioneer and acknowledges the priority of his work, noting that both stud- ies focus on the eight residue classes modulo 30 but follow different directions and methodologies.