BFR more efficient and still it burns more calories?

I can't remember where I've read or heard it, but it is my conviction: when you run barefoot, you run more efficient, but you also burn more calories than when running shod. Is it true? It sounds a bit paradoxical, doesn't it? Efficient per definition means not burning as many calories, right? The more calories you burn, the less efficient you run, all other things being equal? I feel I use less energy when running barefoot, I can run for much longer with less effort -- and yet it also feels like I am using more energy than when I run (or ran) in shoes and go into zombie mode. For instance: when you want to lose weight, should you run barefoot or shod? Does it matter? How does it matter? It feels easier to run barefoot, less like hard work, and yet I have never felt as well trained, my waist has never felt as well trained, as now.
It's a bit of a conundrum, isn't it? Or is it just me?
 
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It's only a conundrum if both bits of information are true, and if you define efficiency as not burning as many calories. Those three things can't all be true together, but since you don't actually know that any of them are true, the most likely solution to the conumdrum is that one of them isn't true. 'Feeling you use less energy' is not going to translate across to a scientific calories in calories out definition since there are such a lot of other factors that might affect your 'feeling' of efficiency. Some studies (as dubious as all the other 'barefoot' studies done by people who don't go barefoot, no doubt) have indicated that running barefoot may not be more 'efficient' in the scientific sense. It seems like a non-question in the real world to me, except possibly for an ultra-runner who needed to carry all his or her calories with them. If the flexion of multiple muscles in running barefoot actually used more calories than plodding a rigid shod foot, would that be a bad thing, or 'inefficient' in a real world sense? Too often, scientific studies, in order to boil down a complex activity to a simple measureable, end up setting up a straw man. CF the many studies that have supposedly looked at the benefits of breast-feeding infants, or low-carbohydrate diets for adults - when looked at, the definition of 'breast feeding' or 'low carbohydrate' is ridiculous to anyone who actually engages in those activities in a knowledgeable way.
 
I've never heard or read that running barefoot burns more calories. I understand we use less oxygen though. I would think we don't burn as many calories as a shod runner because we don't work as hard when we run barefoot.
 
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Sorry, I didn't mean to come over all didactic, if I did. It's just that a little while ago there was some bit of research that came out saying that barefoot running was not necessarily 'more efficient' in a purely theoretical sense and got picked up to imply that there was no advantage to barefoot running, as if anyone ever started it just to obtain some theoretical caloric benefit.
 
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No no, you didn't come across as too didactic, I meant what I said, I like having my beliefs adjusted to correspond to this thing called reality. I wasn't sarcastic. And I'm on your side, I didn't start bfr and do not run bfr because of some theoretical caloric benefit. :)