The Structural Algorithsm for Prime Gap Construction, Analysis, and Infinite Sequential Recurrence

This Prime gap, defined as the differences between consecutive prime numbers, have traditionally been studied using probabilistic models and asymptotic analysis.Although these approaches successfully describe the average behavior of prime gaps, they do not provide a structural mechanism for constructing, analyzing, or predicting the recurrence of individual gaps.This paper introduces a new structural algorithm for the construction and analysis of prime gaps based on the cumulative influence of prime factors.The algorithm generates successive numerical sets by translating an initial set according to the products of the prime factors incorporated at each stage.As the elimination process progresses, composite numbers are systematically identified, producing consecutive composite intervals whose lengths determine the corresponding prime gaps.A central feature of the proposed framework is the concept of Infinite Sequential Recurrence, whereby an initial prime gap generates an infinite sequence of structurally recurring gaps.Once the elimination process reaches a stable periodic state, the recurrence is governed by the repeated product of the prime factors used in the construction.