http://www.polygon.com/2013/11/14/5103898/ps4-what-you-cant-do-offline
Uh oh
Trophy DRM.
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But wait a minute...
Apparently not the day-one update
Uh oh
You can definitely earn Trophies while playing PS4 games offline, but if you try to look at your list when you've disabled the console's internet connection, you'll get the following error message:
[Connect to the Internet] is turned off.
(NW-31484-0)
There's no caching of data, either. Even if you viewed your Trophies and then disconnected from the internet a second later and tried to look at the list again, you'd get that same message. It seems that whenever you bring up the Trophy list, the PS4 attempts to contact Sony's servers, and it won't display the list unless it can sync to get the latest version.
Trophy DRM.
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But wait a minute...
Are people sure that there is no 'offline mode' like the PS3 got?
Looking at the Shacknews video, viewing trophies offline works just fine (video at 2:05):
http://youtu.be/ZhFzDG_1Daw?t=2m5s
Apparently not the day-one update
It is true, though. Just checked.