The phrase "Calorie is a calorie" does not mean what they think it does. It means that each calorie we get must be accounted for. It could not vanish.

Any machine which uses some type of energy to run (input) and produces some type of energy (output) must satisfy this equation:

Input energy = Output energy + Lost Energy

When we apply this to the human body, it should mean that:

Total energy in the food we eat = Energy we spend on all body activities and energy we store as body fat + Energy we lose with breath, sweat, urine and feces.

We can agree that the equation above must apply to all humen no matter which diet they are on.

Now, why do people on low carb diet lose more weight than others when they receive the same amount of calories? The answer is simple, there is more energy lost in their breath, sweat, urine and feces.