The Sherehii Moving Frame Method: A Recursive Additive Algorithm for Prime Number Generation

Abstract

This paper presents a strictly arithmetic, deterministic algorithm for the sequential generation of prime numbers, establishing a structural alternative to traditional analytical frameworks. The proposed method introduces the "Sherehii Moving Frame Method" — a recursive operational mechanism where any target prime is constructed as a localized sum of k preceding primes. By dynamically scaling the number of active vertices k and elastically expanding the outer boundary limits of the frame, the algorithm effectively overcomes wide composite intervals ("sieve traps") without analytical integration, Riemann Zeta function zeroes, or computationally prohibitive factorials. The systemic stability of the generator is bounded by the strict relation P/k > sqrt(P). Detailed manual and computational verifications are provided for critical nodes including the 50th, 100th, 1000th, and 10,000th prime numbers, demonstrating the absolute topological stability of the underlying additive abacus.