The body seems to be able to burn both. What you’re leaving out of the picture is that when the body is assured of enough energy, it increases the metabolic rate and even wastes energy, whereas if you restrict calories, it will respond by lowering the metabolic rate and shutting down the less-essential processes.
Again, this is why Dr. Phinney stresses eating fat to satiety, so that the body is assured of an adequate supply of energy. Do you really think we could actually eat six thousand calories mostly of fat? Of course not; you’d lose interest in your food long before you consumed that many calories.
We are not denying the existence of the laws of thermodynamics here, but the body is a complex system, and how the food we eat gets partitioned is more relevant than the actual number of calories for determining whether we are going to store that food or metabolize it. Gary Taubes cites an early study of the LCHF diet, in which the participants, all obese, were allowed to eat to satiety. They all lost weight at more or less the same rate, but the amount of calories they ate varied considerably among the participants, and one of them, he says, lost weight even while consuming 3000 calories a day.
So it’s definitely a lot more complex than simply saying calories in, calories out. We can’t control our bodies by adjusting our intake or our output; the only way we can exercise control is by altering the quality of what we eat and by altering how we eat in order to work with our natural internal processes instead of against them.