In this theory where does the stored energy (i.e. the fat) come from if not from the calorie intake? How can this be done for more than a short period of time (e.g. I know you can break down muscle for energy) without breaking laws of physics?
This comes up in Hacker News every time a health story comes up, usually quoting a pop-sci writer
Bit of an ad-hom on pop-sci writers there, eh?
Yes. Lean body mass can atrophy from starvation while fat deposits are growing.
The point is more that the equation as stated is a useless way to think about fat loss. Hormones drive the lean vs. fat configuration of your body, not calorie counting. Cut out someone's pancreas and they'll waste away no matter what they eat. Inject testosterone or growth hormone and they will get leaner and more muscular, all else held constant. Inject insulin and they'll get fatter.
Exercise affects your body composition more effectively when you focus on how it changes your hormone profile rather than count calories burned. High resistance major muscle group exercises boost growth hormone and testosterone and increase insulin resistance. Long distance training actually reduces testosterone levels.