As I stopped to take that picture, I was accosted by a security guard. She told me I couldn't be on the grounds without checking in at the office. She was rather adamant about it. Just doing her job, I guess.
So I went up to the office and gave them my "Uh, I was just out for a walk, uh, and I kinda just came in to see what's going on here" spiel. I ended up buying a half-price ticket to see just the gardens.
And I made sure to tell the security guard that when I came out and she asked me what they'd said in the office. Then she warned me to stay on the path. She wasn't unfriendly or anything, though. Just authoritative. A+ for doing her job.
It's lovely out here so far. Lots of shady trees and flowers. Very quiet.