Starting from the orbit regularization framework within elementary num- ber theory, I present a systematic study of the twin prime and related prime gap problems. On the eight independent orbits modulo 30, I prove rigorously that the number of consecutive primes on any single orbit cannot exceed six, with a maximum gap of 150. This bound is guaranteed by the deterministic structure of congruence equations and does not rely on any unproven hypothe- ses. I compare this result in detail with the breakthrough of Zhang Yitang in 2013—the existence of infinitely many pairs of primes differing by less than 70 million, later improved to 246 by the Polymath project. Although the two results differ fundamentally in mathematical nature (deterministic theo- rem vs. existence theorem), method of proof (elementary congruence theory vs. analytic sieve methods), and scope of applicability (single orbit vs. all natural numbers), they share the common conviction that prime distribution possesses deep order. Orbit regularization provides an elementary, rigorously provable exact upper bound (k ≤ 6) for the admissible k-tuple theory of Hardy and Littlewood, and the corresponding gap 150 is strictly smaller than the 246 obtained by the Polymath project, indicating that prime gaps can be further compressed under orbital constraints. The framework can be natu- rally extended to all eight orbits modulo 30, offering a novel perspective on the twin prime problem based on a deterministic mathematical structure.
Huang Feiyue (Wed,) studied this question.