Prime Gear Geometry (PGG): Universal Local Navigation and the RE-PGG Algorithm

This paper presents a paradigm shift in number theory: transitioning from the 'Frontend' abstraction of prime density to the 'Backend' reality of gear-based geometry. By defining the integer axis as an axis gear ($C_1$) and prime factors as periodic interference gears ($C_p$), we demonstrate that prime numbers are deterministic survivors of the axis gear index in a mechanical system. We introduce the Recursive Euler-PGG (RE-PGG) algorithm, proving that primes can be discovered locally at any scale (e.g., $10^{12}$) without processing the preceding number line.