Where's the threshold?

What u mean?

If you don't know what a threshold is then you need a professional.

It’s a transition trim correct?

Nope, it's called a threshold.

Uhm, you don't need a professional. The threshold is essentially the transition piece that comes with exterior doors when you buy them with the frame. They're metal and can't be seen from the inside when the door is closed. My doors have quarter round that matches the rest of the quarter round, I would so something similar and paint it the color of your trim.

"They're metal and can't be seen from the inside when the door is closed"

What is shown below the door currently is the inside shot of the threshold. The threshold or sill is what stops the weather from getting into the house from below the door. You are looking for a reducer strip or the like to make the transition look more finished right? Not really 100% necessary but you can use what ever you like from quarter round as mention above, floor moulding (1/2" x 3/4" typically) which I would use as it looks better, or a small pc of granite or solid surface, etc.

Millworkman- shoe molding in that foyer is 3/4 x 3/4 should I just match it or go with 1/2 x 3/4

What I see is the subfloor with a bit of what looks like thinset or paint on it. I don't see any threshold there. I would be concerned with anyone that has never heard of a threshold DIY'ing.

So, someone who doesn't know the specific name of the metal piece that's in entry doorways can't be trust to trim some quarter round or molding to put front of the door to cover the gap? Uhm, okay. She's asking for trim options, not how to replace a supporting beam, get over yourself.

How about posting a image with the door open for everyone to see what is currently there. You will get a better answer.

It looks like a existing wood threshold that needs refinishing.