If I install a game using a disc, will I be able to download it from the live shop for free?

Please excuse the stream of consciousness question:

So, I just bought Tomb Raider a couple days ago ($20, no biggy, I'm not concerned about the money side).

Anyway, it's on Games With Gold at the moment, but i don't have gold because my house doesnt have internet (i am a caveman).

so obviously, in just over a month, tomb raider will go back to costing money, but since i already have the disc (and may not want to own the disc in the future), will i still get the download for free after the game is no longer free?

So, more specifically, when i do get internet connection in however many months, and just wanna run around the map on Tomb Raider, or maybe play it again on hard mode or online (god forbid) will be able to do that?

TLDR: if i own the disc, but just want the game on the HDD, do i have to pay for that?