The app is correct.

You're base calories should already take into account the deficit. (ie. you would normally need 2200 calories to maintain your weight, but to lose ~1lb/week your base calories would be 1700) If you burn 1000 calories with exercise, then you should increase your total intake for that day by 1000.

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Strava

If the 1700 base calculation is correct for the target weight loss, then yes you can eat the full 1000 that you burned and still lose weight.

It might seem extravagant, but it's not. Try going without that 1000 for 2-3 days in a row, and you'll literally be thinking about food nonstop, all day long since you'll be in such a deficit.

The real challenge is that when you start replacing those missing 1000 calories + base calories, is making yourself STOP EATING at the calculated number. It's not that hard once, mildly hard for a week, but surprisingly challenging to do consistently for a month!

While I don't use the app you mentioned I know that my daily limit is about 1500 calories to lose 1 lb every two weeks.

If I don't eat then I'll be weak the next day and won't be able to complete workouts as planned.

To a degree. The body is an amazing thing.

I also use LoseIt, and it was bang-on from about 175lb. to 163lb. Then at 163lb I plateaued. So I kept ratcheting down the base calories. Still a plateau. After several weeks of ratcheting down I realized I was walking around in my 75-degree house bundled up in fleece. My metabolic rate was just cratered - though my workout performances were still pretty solid.

Though this was around ~10% bodyfat for a 47 y.o. male, so I'm calling it my physiological "floor" for maintainable weight. I've actually gone up to 166 just to be a little happier (and warmer).

So my n=1 is that base metabolic rate can vary a ton. I believe this is corroborated by some recent research that suggests the body can adapt pretty darn quickly to calorie restriction, so you have to be careful using static numbers.

Edit: I realize this answers a different question than the OP was asking...just pointing it out.