Correct, brown wire bypasses the ballast during cranking saving the low voltage for a stronger start up, then when ign switch returns to RUN, the brown wire is not anymore feeded by ign switch BUT keeps linked to the ign circuit through the ballast buddy!! So KEEPS HOT unless you disconect it from ballast, which is not posible because is spliced into the same coil wire and if you disconect it, will stall the engine

TEST IT! Start up the engine and test for voltage at ign switch brown wire and will get a low voltage power there. Is te same power the coil is getting!

I can't believe this si being really discussed! Its so clear!