According to this Golf Digest article, Bryson needs a miracle today if he wants to get Special Temporary Membership. Since the article says it well, I'll just quote this one paragraph:
The Travelers marks DeChambeau's 12th start on the PGA Tour this season. By rule, non-tour members are allowed just 12 appearances per year. Theoretically, DeChambeau -- if he musters a comeback Friday and somehow finishes second or third in the tournament -- could earn Special Temporary Membership, which could bestow invitations to the John Deere Classic and Wyndham Championship. That route is highly unlikely.The article then details the various options Bryson has if he can't get it done today... but it certainly looks as if Bryson's got a lot more work ahead of him, no matter how it all plays out.
Perhaps the biggest irony here is that former amateur John Rahm, who turned pro just a couple of months ago, locked up his Special Temporary Membership last week. (You can read the details of that in this Golfweek article.) Rahm gained his status in just FIVE events, highlighted by a T3 at the Quicken Loans Invitational and a T2 at the RBC Canadian Open.
Sounds to me like somebody's got bragging rights!
If you want to see if Bryson can pull this off -- he's got to make the cut to have any chance, and he's currently +2 (T113, 8 strokes off the lead) -- he tees off at 2:15pm ET today, which will be well within GC's broadcast window.
Can the scientific approach to golf create a miracle? That remains to be seen... but it should certainly be an interesting experiment to watch.