This manual might help you.
http://dl.dropboxusercontent.com/u/12502267/B&S Service Manuals/05_273521TwinCylinderOHV.pdf
It sounds like you have an electric pto clutch, and you charging system is not working, and it I running your battery down after
a time, and it shuts down because the fuel solenoid shuts it off, and will not start again until the battery is recharged.
If you don't have an electric pto, then something is putting a big drain on your system, but the first thing to check would be
ac voltage coming from the alt., and also checking the voltage from the regulator {dc}.
It shows a switch on the oil sump blowup, but doesn't have a part number corresponding with it, which means that your engine doesn't have one.
He said it shuts off when the battery runs down totally, which is why I think that he should address that problem first, as the pto switch is draining his battery, and the fuel shutoff solenoid is likely shutting it down. I agree, pulling the kill wire to the coils would help narrow things down a bit, but I doubt that his coils [both] are the problem, as having both of them die at the same moment each time is very, very unlikely. Testing the charging circuit is extremely simple. I think he will find his trouble there. free and easy, that is always my next choice, instead of wasting money.
youre saying since your wife is light enough to trigger the seat switch,does this occur when you mow also?..if you let this mower run on a flat surface does this problem still happen?dont go replacing relays just yet,lets investigate a little further
Is your PTO on when the tractor shuts off? [/QUOTE
Not always sometimes the PTO would be on when it cuts out but often it is off ie: the PTO button on the control deck is down and the PTO is off -but if I am right the switch on the PTO unit may still be reading as On.

If it likes to cut out/die on slopes/QUOTE]
Fish it does NOT like to cut out on slopes. All of my posts I have said that " engine cuts ( which may be legitimate)" in other words it may be my wife moving her cute little a... too much and triggering the cut out OR we may stop and turn the engine off. The problem is NOT the engine stopping - IT IS NOT the problem. The problem is that after it has stopped ( for whatever reason ) on many occasions it will not then restart.
The conundrum is what fault can let the engine run with no issues ( and I do mean no issues ) - but will not let it re-start .... it is only re-starts. Normal cold starts are perfect.
If not, can I borrow her?????