You are absolutely correct. It uses the front narrow band sensor to determine the actual AF ratio and then tweaks the long and short fuel trims to get to whatever it is shooting for. As soon as you see closed loop for the fuel system it’s using it.
When the sensor fails it reverts to open loop and runs on fixed parameters to determine injector pulse width based on TPS, MAF, RPM, ECT, IAT. It try’s to run around 12:1 AFR to prevent a lean condition and be drivable.
On a carbrated vehicle or motorcycle the O2 doesn’t swing rich and lean so it’s much easier to use. (Just not as accurate) go wide open throttle and watch your front sensor voltage. It will instantly stop swinging high low and settle in between .88v and .95v because it goes open loop. If your engine and fuel system is ok anyway. It shoots for 12:1 at WOT in open loop but know way to know if it made it. I will watch my LTFT to make sure it’s near 0 and I’ll also do a WOT acceleration watching the voltage of the front sensor to make sure my pump and filter are flowing properly.
Fuel injection swings rich and lean on purpose. One of them is to provide oxygen to the catalyst in the converter to burn the unburned hydrocarbons. The catalyst absorbs it and then gives it off. With out it you would need air injection at all times like the old days when engines had belt driven “smog” pumps pumping air into the exhaust manifold and into the CAT around halfway back the honeycomb.
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