Yes, the software runs 100% on your PC, other than checking to see if the license is still properly paid for via the internet, which it does once a month. You can disconnect from the Internet and it will run without issue for a month.
A quick Google reveals this authoritative FAQ answer:
Do I need to be online to access my desktop apps?
No, the desktop applications in Creative Cloud, such as Photoshop and Illustrator, are installed directly on your computer, so you don’t need an ongoing Internet connection to use them.
An Internet connection is required the first time you install and license your apps, but you can use the apps in offline mode with a valid software license. The desktop apps will attempt to validate your software licenses every 30 days.
Annual members can use the apps for up to 99 days in offline mode. Month-to-month members can use the software for up to 30 days in offline mode.
The LR Help command takes you immediately to the Web site, but I find the Help command pretty useless. I downloaded the PDF manual to my computer.
You need an active Internet connection to use LR Mobile, of course.
That's good news. Thank you both.