I'm not questioning whether ketosis is a real thing. I'm asking if you know of a study of the Atkins diet which keeps track of the calories being ingested and thereby demonstrates that people are ingesting more calories than would otherwise allow them to lose weight. You're flat-out wrong that that's what my link said, as it explicitly says (and I have already quoted) that when they looked at how many calories people were ingesting that those on the Atkins diet were eating the same amount as those people on calorie-controlled diets, such as Weight Watchers.
As for 22 calories, you seem to have misunderstood the relevance of that. Yes, that's calories the participants wouldn't have lost on other diets. But, seeing as that equates to ingesting half an apple, and can be burnt off by walking at a normal pace for 6 minutes, it's negligible and cannot be the primary method of weight loss.
So that's what you're looking for - a study which establishes that people on the Atkins diet actually do eat the same or more calories as they did when they gained or maintained their weight. And if such a study hasn't been done - not even by Dr. Atkins himself - then it might be worth asking yourself why not. The first thing to do in science is to determine whether or not an effect actually exists, before you go trying to explain how it works.
As I said, the studies are in the book. I linked the book above, it includes preview and there are many papers available for you to track (good luck tracking papers from the seventies). I don't need what you're asking for. I had my own case study on my own. We documented it with my father (a gastroenterologist and diabetologist). I never estimated the overall effect -and it's almost impossible to do, but I have firm hypothesis on that). I lost 22.8 kg (let's divide it in 3 kg muscle, 1.8 kg water and 18 kg fat tissues) to total some 140,000 Kcal. My intake was during the 59 days of the diet averaged 1350/1400 kcal. So my "apparent consumption was 3700 kcal/day.
By those times my ideal weight was 79 kg and my activities, time of the year and age determined some 30/31 Kcal/kg, that is, 2400 Kcal/day. I walked a lot during the diet, with peaks of 14 km some spare day and no less than 6km walked each day. Let's say an average of 8 km a day. With my ideal weight and my average real weight in that period, plus I walked very fast (some 7.5 km/hour in the beginning), that means my walking -and the extra muscle work during hours after each walk-, I burnt some 700 kcal a day. That leaves at least 600 Kcal a day to be eliminated as ketones. That means no less than 80 g ketones a day. Let's suppose 30 of them go out with urine. I drank like a camel in the oasis and pee like an elephant, let's say one gallon a day (now it's normal to me because of the diuretics). That is a concentration of 70mg/10 cm3 for ketone, that is IN THE PURPLE, EXACTLY AS REQUIRED BY ATKINS.
Don't take me wrong:
You lose a lot of weight
You lose you health while you do so: you gave up your bones, your teeth, your immune system weakens, etc.
I never repeated it, and not for lack of courage and will.
There's also a trap in losing weight so easily: you eat inconvenient food as you'll wipe the fat away later with you super diet. That's the worst mistake a person trying to lose weight can do. Anyone can lose tons of weight, not many can become slim people for ever ... and that's the point.
Today, to me Atkin's inspired diets are just a reinforcement just in those moments your body becomes a cheap expender and you're hungry even when you've just eaten. The "ketogenic" factor gives the diet a boost. It diminishes appetite just as a side effect of ketone. It avoids your body burning little energy. But the minute you start to lose weight steadily you have to abandon the ketogenic factor and draw back to the safe and sound diet.
If any of you wants to know what I eat, avoid eating or plan to eat during my current diet, just ask.