Trapped-Ion Page-Wootters Experiment: Protocol for Testing Ultrametricity

Abstract

We propose a trapped-ion experiment to test the Sufficient Condition Theorem: that diagonal clock-rest coupling produces exact ultrametric conditional state overlaps (UVR = 0%), while nondiagonal coupling produces a universal ~32% violation rate. All required capabilities are established with current trapped-ion technology. Estimated timeline: 8 weeks on an existing apparatus.