I haven't been able to test yet, but I bet that when trying to kill the paradox's past, it will only choose pasts that you've defeated before, rather than other gungeoneers' pasts. - Unfortunately, there is no way to get The Bullet as The Paradox, or even get to the chest that holds the gun. The litch will grab you as soon as you step on the stair with the hell entrance with no way around.
Excluded starting items
I'm noting this down here for future reference. As of V2.1.7h1, the paradox cannot start with these items:
- Lich's Eye Bullets
- Gungeon Blueprint
- Cartographer's Ring
- Finished Gun
- Bullet That Can Kill The Past
- Old Crest
Additionally, as of V2.1.9 (or maybe 2.1.8) the second, non-starter gun can not be the Blasphemy.
Cosmic rift conditions
As of V2.1.7h1, all of these conditions must apply for the game to spawn a rift when generating a floor:
- It's the Gungeon Proper, Black Powder Mine or Hollow
- The player has beaten any past from amongst The Pilot, The Convict, The Marine, The Hunter, The Robot or The Bullet
- The player has not yet unlocked the Paradox
- The player does not currently have the paradox effect from a cosmic rift
- Pass a random test with a 20% chance of success
it says the paradox can start with non-unlocked starter guns, but I've played a few dozen runs without getting the Slinger, and I /do/ have the gunslinger unlocked. This seems like he can't have it, but it could just be rng - has anyone started with the Slinger?