noninvertibility

Return Is an Update: Memory, Feedback, and the Next Ground v0.1

The FRC toroidal qualifier means that outward passage can return into a later ground, not that the path is a geometric circle. This paper models a state-and-environment update and separates memory, feedback, delay, and environmental input. If a selective threshold merges two reachable states and every downstream step is deterministic under fixed inputs, the completed cycle cannot distinguish their histories. Its return map is noninjective on those histories. The next ground need not equal the prior ground; equality requires a separate recurrence test.