The concept of a closed timelike curve (CTC) is still a topic of theoretical debate and has not been proven to exist in reality. Therefore, it is not possible for an observer to prove they are in a CTC as it has not been observed or proven to exist.
If CTCs do exist, an observer would not be able to physically detect or measure their presence. This is because CTCs involve time travel, which would require a violation of known physical laws and the concept of causality.
Currently, there are no known experiments that could definitively prove the existence of CTCs. However, there are theoretical models and thought experiments that explore the possibility of CTCs and their potential consequences.
If CTCs do exist, it is theoretically possible for an object or information to travel through them. However, the concept of causality and the potential for paradoxes make it a highly debated and controversial topic in the scientific community.
If CTCs were proven to exist, it could completely change our understanding of time and the laws of physics. It could also have significant implications for concepts such as free will and the possibility of altering the past. However, this is still a highly theoretical and speculative topic with no concrete evidence or proof at this time.