The most appropriate one-word substitution for the given group of words "An act of carefully watching someone or something especially in order to prevent or detect a crime" is "Surveillance."

Explanation of the options:

☀ Thearchy: Thearchy refers to a government or rule by a deity or deities and is unrelated to the act of carefully watching to prevent or detect a crime.

☀ Decanter: A decanter is a vessel used for pouring or serving drinks and does not relate to the act of surveillance.

☀ Survey: While "survey" can involve observing or examining something, it is a broader term and does not specifically imply the vigilance associated with preventing or detecting crimes.

☀ Surveillance: This word accurately captures the act of carefully monitoring or watching someone or something to prevent or detect a crime. It aligns with the given group of words and denotes the specific context of observing for security or investigative purposes.