You want to lose fat and gain muscle at the same time and everyone says it is impossible

Everyone tells you body recomposition is impossible. Give it your weight, height, age and training, and it calculates your TDEE, then sets out the recomp strategy: maintenance calories, high protein, the exact daily targets for protein, carbs and fat, and why this approach works, plus how to eat differently on training days.

I want to do a body recomposition: lose fat and build muscle simultaneously. I am {WEIGHT}, {HEIGHT}, {AGE}, training {TRAINING_DAYS} per week with {TRAINING_TYPE}. Calculate my TDEE, then explain the recomposition macro strategy: maintenance calories with high protein, showing the exact numbers and why this works. Give me daily targets for protein, carbs and fat, plus guidance on whether to eat differently on training days versus rest days. Show all calculations so I can follow the logic.