on 331 cals a day you would be starving because you are literally starving in the true definition of the term. 1200 a day is the min recommneded to get all you vitamins and minerals.
the main secret to atkins is it is a low calorie diet. a ketone is a partially burnt fat molecule. a gram of fat is around 9 calories but a ketones as a partially burnt fat means that your not getting 9 calories off it. so you can eat 1500 a day but you wont burn that much.
its like serving a plate of 600 calories but leaving some on the plate when you finish. what you put in isn't what you get
To infinity and beyond!