This comes up in Hacker News every time a health story comes up, usually quoting a pop-sci writer. You've stated it by far the clearest with no vagueness or handwaving so hopefully you can answer this (or provide pointers to research):
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?
It's better to argue "weight gain = calories in - calories out" is not the whole story than to argue that it's completely wrong - you can't eat and drink nothing for a week and just get heavier, but what you can do is more like... eat 3500 calories extra in a week without putting on a pound of fat (1lb of fat ~=3500 calories) because your body is not trying to store as much as it can as fat, or eat -3500 calories in a week and end up slowing down your metabolism and moving less and destructing other bodily parts so you turn it into a calorie surfeit and do put on a pound of fat.
This comes up in Hacker News every time a health story comes up, usually quoting a pop-sci writer
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.
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?