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Yeah but you're not a trained engineer after years at Starfleet Academy and years of postings who is currently on the Flagship. If they had the shields up constantly around it, it might imply that they don't trust their crew. Starfleet is all about trust and working together. If they shield the core constantly they're not trusting the engineers they trained to work on it and fix it. Imagine you spend years at the Academy studying the best in warp mechanics and what have you. You finally get to serve and after years and years you find that no one is trusted. That despite all your training on how to deal with it, you'd need to apply for permission to gain access to the thing your department is dedicated to. Those feelings of being untrusted or babied can really get to you after a while.