I run off for a little GAL activity and all heck breaks loose on your thread
I think everyone is saying the same thing and saying it well. I just want to embellish a few points.
Your wife is trying to sell you a load of goods that this arrangement is "special", "adult" and a presumably civilized.
That is a load of hooey.
What's really happening is that your wife is getting away with whatever she wants - comfortable living arrangement and whatever time she wants with OM. She can tell you it's over all she wants - and it may be true for her- but when you end a relationship the "adult way", you do so with dignity, integrity and respect for the other parties feelings - you don't just expect them to eat whatever chit you are dealing out (sorry for the salty language).
Telling you its better for the boys? Ma-lar-key. Period. Full Stop. Having them stay in an environment where their father is disrespected on a daily basis by her actions, is not setting an "adult" example for them, unless she wants to raise timid creatures who let others walk all over them.
And while I have myself roused into a fit of righteous indignation - I'm just going to get a little judgy and say that as a mother, I find the idea of conducting a full blown affair while my child is facing major surgery particularly.....gross.
One last thing OD - I think most of us who have a WAS with an AP would like to see the relationship end. You know what ends those relationships? Having to deal with actual consequences.