The ketones spilling over into your urine come from a combination of dietary fat and body fat so, yes, it is possible that all you are seeing is dietary fat but, as it turns out, this is rather unlikely. Read your Atkins book in the chapter on the metabolic advantage. There's also some discussion of this topic in the Low Carb Tips link above.

The proof is in the pudding. Give it an honest and faithful try and see what happens for YOU.

Cheers!