Small Engines (Lawn Mowers, etc.): craftsman riding mower, craftsman riding mower, scematics


Question
I am having trouble with a craftsman 19.5 horse briggs v twin 42" cut riding mower.  Model #917.259561.  The problem is that it will start as usual and run for 10-15 minutes and then it will start dying as if it is starving for gas.  If I disengage the blades and full choke it I can usually limp it about 50 yards before it completely dies.  If I then let it cool for about 30 minutes it will usually start back up and I can finish mowing the yard (about 40 minutes) and it will run fine.  That is until recently, now it won't cooperate unless I let it sit for hours.  I will throw this in though, the last time I mowed it did the entire yard without even a sputter but today it pulled its old tricks again.
I won't bore you with my thought processes on all of this but I will list all of the things that I have fixed/replaced so far:
replaced Air filter
replaced fuel filter
replaced plugs
blown out the fuel lines
replaced fuel pump
checked muffler for clogs
broke down the carb and cleaned all jets and float
cleaned all the connections on the seat kill switch

None of these things have changed how the mower reacts at all.  Someone told me to check the coil but when I checed the electrical scematics at sears.com I saw nothing that looked like a coil at all.  Please let me know what direction you think I should go next.

Answer
I know you said you put a new air filter on, but for testing purposes ONLY, please remove the air filter and try it.  Does the problem go away?  

How fresh is your fuel?  Is it fresh (within the last week) from the gas pump?  To rule out if its a fuel problem, find an old fuel tank off of a push mower and hook a fuel line from your carb to the tank and try this.  This is called a remote fuel tank and will quickly tell if its a fuel related problem or not.

Let me know