This is extremely simple minded almost to the point of being dogmatic.
As someone who has dieted all my life and *finally* gained control of my weight through lo-carb, I can by experience tell you that it just AIN'T SO SIMPLE AS THAT.
I realize that you got a warm fuzzy feeling when you took your physics class in University that you were starting to discover a UNIVERSAL TRUTH that you could apply everywhere and viola! you understand how things work.
BS - human metabolism is EXTREMELY tricky, and is constantly changing its energy expenditure based on dozens of obscure variables.
It was perhaps unfortunate that the unit of energy was called the 'calorie' which of course is the same name, but by no means the same UNIT used to raise water temperature. This allows the same mechanistic mindset to spout the meaningless drivel you have uttered.
The human body is an exquisitely crafted FEEDBACK SYSTEM, not a big beaker full of distilled water.
"A calorie is a calorie is a calorie."
On the surface, this is as sensible as "A is A".
A gram of fat is about ten calories, a gram of carb about four. So, it is reasonable to think--"eat your grams in carbs and you'll eat fewer calories. Eat fat and gain."
It takes for granted that the body treats all calories the same.
I keep hens, and it is remarkable to me that I load them up with corn and some calcium supplements, and they give me eggs (fat and protein)wrapped in a thin layer of stone. This is a pretty good indicator that the living creature is an astonishing factory of chemicals, and what you put in is certainly not what you get. If I want to fatten a hen, I don't feed it cheese and meat and oil. I feed it corn. Now, even grain has a little fat and protein in it, but not as much as I "put in" to the chicken to receive an egg.
I don't push Atkins--I need someone to bake cakes and bread and cookies for. My family eats sweets (though probably not as many) and I load up a salad with creamy dressing. Or have a bigger portion of steak. Or put butter on my broccoli, something I never did before. I consume more actual calories now than when I was eating a "normal" diet because I snack on nuts, macadamias by the handful!, instead of crackers and pretzels.
It's not protein vs. carbs, it's protein and fat vs. carbs. Atkins won't work without the generous addition of all sorts of fats. When I indulge in popcorn, I pour on some butter because it keeps me from eating too much popcorn. The fat also makes it harder for the body to "get at" the carbs.
Fat is more interesting to me as a means of weight loss than protein, which also gets stored as fat if you eat enough of it. Fat will kill your appetite quickly, keep you satisfied. I read that Russian soldiers in Siberia get a ration of butter, which they eat like candy bars. This generates heat, I read.