Update time, I wasn't able to get out of work to go to my mechanic Thursday. By the time I left, 14 hours later, the codes had reset.
Next day I started over with the relearn procedure and drove straight to my mechanic. Never turned off the car and he took it in and tested it. Finally got a pass. All monitors ready with no errors. He turned the car off, turned it on 5 min later and all was well. As I'm driving away, the "tighten gas cap" message pops up. I parked the car for about 8 hours, get back in and all monitors are reset again.
The good news I now have my inspection so I can resume working on this when I have more time. Bad news I'm still not sure what's going on.
My best guess for the resent is some sort of power failure. Now the car is not starting rough, hesitating, or anything noticeable. Could be issues with Honda's charging system, so I might pull some mods in the spring and take it to the dealer.
Has anyone come across a fix for the gas cap message? There's no issues with the cap and the EVAP monitors don't show errors other than that light. I know with past accords and such, there was a false positive in the system so I've never given it too much merit.