Modular Recurrence and Periodic Structure in Padovan, Fibonacci, Lucas, and Jacobsthal Sequences: An Analysis of Prakriti and Avyakta Dynamics

This paper presents a mathematical analysis of recurrence, modular periodicity, and digital-root dynamics in several classical integer sequences, with special attention to Padovan, Fibonacci, Lucas, and Jacobsthal systems. The study compares two complementary structural regimes, referred to here as Prakriti and Avyakta, not as metaphysical categories but as analytical labels for two distinct recurrence patterns: a stabilising 3-6-9 framework and a complementary 1-2-4-5-7-8 flow. Using Pisano periods, modular reduction, and sequence alignment, the paper identifies recurring phase shifts, reset points, and finite-cycle behaviours, including the Padovan period of 114 terms. The notion of “reset” is treated as a structural transition within recursive systems, rather than as a mystical or numerological claim. The result is a comparative framework for understanding how distinct integer sequences generate periodic, offset, and alternating patterns across modular arithmetic.