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I have two identical iPhone 4g phones, one with a broken screen (my current phone) and one with a perfect screen from my past carrier. Can I swap my current motherboard into the other phone without compromising the data?

You first need to make a backup of your iPhones applications and the apps data within iTunes or iCloud and you may need to manually copy off some of your data files as well if you used your phone as storage to transfer files between your PC’s. Here’s a good writeup on the needed steps using iTunes Transferring Apps to a new iPhone Here’s a YouTube on how to do it Transferring via iCloud Of course you’ll need to pay Apple for this service.

There is no loss of data when swapping boards. Just swap them, everything will be there when done.

I understand your goal is to retain old motherboard into a new device. Yes. All information is contained on the chips on the motherboard. While icloud backup is highly recommended—because any mishap could destroy your motherboard circuit or cause memory erasure—if backup is impossible, swapping board (or more accurately, swapping everything except the board) should allow you to regain full phone function. (THEN immediately do a backup to prevent the situation in future.) Be sure exact same phone models. This trick may not work with newer models. We assume no liability if you brick your own phone in this process.

No, you cannot because everything related to memory is built in on board its not like a PC where data is stored in different components like hard drive and ram etc. Its better to back up you whole data first and the restore it in new board.