I've always been confused by the "calories in, calories out" claim. It is usually used to mean that weight gain/loss is proportional to calorie surplus/deficit.

I've always been confused by the "calories in, calories out" claim. It is usually used to mean that weight gain/loss is proportional to calorie surplus/deficit. But I don't understand how that is a restatement of the first law of thermodynamics. What I could believe is the claim that the number of calories stored in a person's body is proportional to their lifetime caloric surplus, from which it follows that the change in the number of calories stored on a person's body over a period of time is proportional to their caloric surplus over that period. But weight is not proportional to stored calories, since fat is more calorie dense than protein. So in order to gain muscle, lose fat, and maintain the same weight, doesn't a person have to run a caloric deficit?