Kohler 16 hp cast iron 341 aqs. Starts and idles beautifully. TOUCH the throttle and the engine starts bucking like it's coming off the tractor Back off throttle and back to smooth idle. help please!

if it is a petrol engine then the governor arm will hold the carby open when stopped
as the engine starts the governor moves the arm to close the carby off to idle. the speed setting is by a spring attached to the throttle level and a point on the governor lever

May be the throttle position sensor, or the idle air control valve

bad bearings on crank or worn oil pump. Pressure will be up when cold, but will drop when engine warms up. Check for knock in lower end. If so, bad crank bearings.

Most low idle speed problems on these engines are due to a faulty idle air control valve but since you state the idle will go up and return to idle then i would say it is working properly i think you have a throttle body coking problem ( dirt and grit accumulates around the butterfly valve in the throttlebody ) to fix this simply remove the fresh air hose from the throttlebody and open the throttle all the way up use a clen rag and wipe clean you may use a solvent such as B12 on the rag but dont use too much

closes bet its a bad alternator, bad battery or loose connections, last 2 options throtle needs to reprogram,bad as well.

Check you tps sensor sure that's your problem

IF IT IS HOT AT IDLE IT IS LEAKING, JUST THE ENGINE AND LET IT IDLE, WET YOUR FINGERS AND TOUCH THE CAST IRON PART OF THE VALVE NEAR THE BASE.

Remover your IAC motor (idle air control) on throttle body and clean it well, especially the pintle, (pointed part) and hole (port) it came out of with throttlebody spray cleaner, then re-install & try it. If it's good for a while, then starts to do it again, replace the IAC. Quite often this will cause stalling, low idle etc, but will not store any trouble codes.

Have you checked the fuel pressure while pressing the gas pedal? Pressure should stay above 40PSI

A faulty gas cap can cause the sensor to come on.
Clean out the throttle body with a cleaner and see if the idle improves.