The Universal Pisano Period of Second-Order Linear Recurrences in the Modulo-9 Ring

Update (August 2026): This version includes new sections detailing the Empirical Quantum Verifications simulated via VQE, QPE, and Hamiltonian Dynamics on Qiskit/IBM Quantum architectures. Source codes are open-source.We generalize the Pisano period phenomenon to all second-order linear recurrence relations. Through computational brute-force and algebraic ring theory, we demonstrate that any sequence of this form fundamentally locks into a Pisano period of 24 in modulo-9 topology. Formal Verification & Source Code: The rigorous Lean 4 proofs and framework codebase are open-source and available at the Ouroboros GitHub Repository.