So my son is now just over four months old. He was 3 weeks and 2 days early and about 5.5lbs when he was born and he's now almost 12 so he has doubled his birth weight and then a little more in four months.

I've been exclusively pumping since the very beginning so I've always known exactly how much he's eating. At his four month check up my doctor stated that I need to start supplementing with formula as he hasn't crossed the 5th percentile line (he has very consistently tracked at the 3rd percentile for weight since he was born, and was tracking on the 2nd percentile while I was pregnant). Now because I know how much he is eating, I questioned this a bit because I literally cannot get him to eat any more. When he eats a bottle he almost always leaves a bit behind and gives clear signs that he is done (usually pushing it out with his tongue). If he finishes the whole thing I give him more until he shows he is done. My doctor didn't give me a satisfying answer as to why I should be supplementing with formula when I have more than enough breast milk for him. Essentially just telling me that it will help him put on weight without giving me any reasons why. I am a very logic based person so I need clear reasons to help me understand why I should be doing something. I have a chest freezer that's basically full of frozen milk right now and even as I'm slowly reducing my number of pumps per day I'm still managing to freeze at least one session while having enough to feed him throughout the day. So I can't justify paying for formula when everything I've read says breast milk is better for baby when available

Has anyone else experienced this? Is there a reason that my doctor has suggested this that she didn't make clear to me even when I'm asking for clarification? Everything I can find online seems like formula supplementing is for when baby isn't getting enough/mom's production can't quite keep up. Any insight anyone else may have from their experience would be greatly appreciated! I want to do what's best for baby but need some help seeing the potential reasons why formula may be necessary here.