Topological Properties and Pisano Period of a Modulo 9 Sum-and-Shift Recurrence Matrix

The 99-State Pisano Root-Shift Matrix is a closed-loop Finite State Machine (FSM) operating on the domain of two-digit integers (01 to 99). By utilizing a specific ”Sum-and-Shift” recurrence relation utilizing digital roots, every number forms a perfect bijection. The entire system resolves into five isolated topological loops, governed strictly by the Pisano period π(9) = 24. This paper documents the derivation of these sequences, formalizes the system using linear algebra, establishes its statistical constants, and explores its revolutionary applications in mechanical kinematics and zero-storage distributed computing.