Please capitalize where necessary: however, once the the pivotal moment occurs the film slows right down and begins to lag as the aftermath of jack’s actions plays out. ted levine is introduced as lewis, another part of the criminal gang jack has crossed, and adds a little bit of off-kilter menace but the story seems to fall back on convention as jack mooches around trying to figure out his next move.
A: However, once the the pivotal moment occurs the film slows right down and begins to lag as the aftermath of Jack’s actions plays out. Ted Levine is introduced as Lewis, another part of the criminal gang Jack has crossed, and adds a little bit of off-kilter menace but the story seems to fall back on convention as Jack mooches around trying to figure out his next move.

Please capitalize where necessary: we've developed a parallel theory ... of computation that deals with the real, physical world.
A: We've developed a parallel theory ... of computation that deals with the real, physical world.

Please capitalize where necessary: mathias has spent the day going through the medical supplies, while the rest of us have enjoyed the task of finally being able to clear away the cargo clutter, and stock cargo in the garages.
A: Mathias has spent the day going through the medical supplies, while the rest of us have enjoyed the task of finally being able to clear away the cargo clutter, and stock cargo in the garages.

Please capitalize where necessary: in the slain in the spirit counterfeit, they fall backwards, either in response to the wave of the speaker's arm or as a result of a church leader's touch (or push in some cases).
A:
In the slain in the Spirit counterfeit, they fall backwards, either in response to the wave of the speaker's arm or as a result of a church leader's touch (or push in some cases).