Periodic Pairing Matrix Computation Model: Theory, Algorithms, and Applications

This paper introduces a computational framework called the Periodic Pairing Matrix model, which provides a foundation for modeling periodic interactions between cyclic sequences. The model formalizes pairing between two sequences using configurable step sizes, creating a deterministic pattern that repeats over a fixed cycle. We present a comprehensive analysis of its properties, including periodic behavior, coverage conditions, and vacancy patterns. The framework offers a unified approach to resource scheduling, sparse neural network design, and other domains that benefit from structured periodic interactions. Through case studies in distributed computing and deep learning, we show improvements in resource utilization, reaching full utilization under optimal settings, and faster neural network training with gains of about forty percent, while maintaining performance. The Periodic Pairing Matrix model enables principled design of systems with predictable and analyzable periodic behavior.