Mathematical analysis demonstrates a single-rule solution to the parity problem in one-dimensional cellular automata, confirming distributed consensus without global communication.
Key Points
To formulate and formally prove a single-rule one-dimensional cellular automaton that solves the parity problem on odd-sized cyclic configurations.
Analyzed the single-rule BFO formulation and its known computational edge-case failure modes.
Constructed an updated local transition rule for one-dimensional cyclic binary lattices.
Developed a complete mathematical proof establishing convergence to homogeneous fixed points based on parity.
Rectified the specific configuration failure present in the original BFO rule.
Delivered a rigorous, complete mathematical proof confirming that the single rule reliably drives odd-sized cyclic lattices to all-0s for even parity and all-1s for odd parity.