#### Zero-Error Sum Modulo Two with a Common Observation

This paper investigates the classical modulo two sum problem in source coding, but with a common observation: a transmitter observes $(X,Z)$, the other transmitter observes $(Y,Z)$, and the receiver wants to compute $X \oplus Y$ without error. Through a coupling argument, this paper establishes a new lower bound on the sum-rate when $X-Z-Y$ forms a Markov chain.