So, after some further thought, I think that I actually agree with a lot of what Menno proposes - just not his reasoning for getting their. Frankly, as stated in the post above, his math is often wrong, his logic has flaws, and he has contradictions throughout his writing that he either isn't aware of, or doesn't care to fix.
Having said that, I agree that you can gain muscle while losing fat, and gain fat while losing muscle, and and thus lose weight while gaining fat, etc. At least in theory.
What I take exception with, especially since he keeps trying to sound smart by invoking the laws of thermodynamics, is the idea that you can gain weight in a deficit - other than water weight. I mean, sure I can gain weight in a deficit - I can drink more water, I can put on a shirt and pants and some shoes, I can carry a backpack.
Upon reviewing his website, he constantly confounds water weight with muscle weight. He either doesn't grasp the difference in the situations he uses, or he is just being disingenuous. Or both.
I CAN mathematically come up with a situation where you can, theoretically lose weight in a deficit; which is what he is proposing - and that is that a pound of fat has 3500 calories and 15% water, while a pound of muscle is closer to 75-80% water and about 15-20% protein. So, only accounting for calories and a direct 100% conversion, you would require, let's say 90 grams of protein or 360 calories worth of protein + water to gain a pound of muscle and a pound of fat could thus provide the caloric need to build around 9 pounds of muscle in that scenario.
So, on the really high level, you could gain 9 pounds of muscle while losing 1 pound of fat - for a net gain of 8 pounds - if you were eucaloric and you drank sufficient water.
However, as Menno points out in other areas, you would have to get the nitrogen in order to build the protein from protein - you can't get it from fat.
So lets say you burn 2500 calories/day, eat 2400 calories per day and 1800 of those calories (75%) are from protein - that's 450 grams of protein per day.
If, for some reason, you could get your body to use more than typical of that 450 grams of protein for actual protein synthesis (like steroids) - and break down fat in order to make up for the caloric deficit and to compensate for the protein sparing - then maybe the math itself can play out (although Menno's math is still way off and just wrong).
In other words, if I actually use, say, 300 grams of the 450 grams for protein synthesis - and replace the 1200 calories of protein with 1200 calories of stored bodyfat - then I will lose 1/3 of a pound of fat roughly, and likely have enough actual material to build a pound of muscle if you add in 0.8 pounds of water. The net of this will be that I've gained 0.66 pounds and was in a 100 calorie deficit. Yes, I gained weight in a deficit. The impossible is possible.
Except, there are a number of issues with this. One is, the synthesis of protein requires an enormous amount of energy. Just maintaining your nitrogen balance likely accounts for 20-25% or more of your daily TDEE. So you would be increasing your TDEE just by building the protein, which would mean you would need to burn more fat...which would be more weight lost so that would bring the increase your actual deficit and also reduce the amount of weight gained. If you could even trick your body into doing this.
Further, the energy costs of mobilizing body fat need to be accounted for, which would further increase your deficit and decrease the weight gain.
Another issue that comes to mind is that, if you're in a deficit, ATP levels will be lower - and this increases AMPK and this signaling makes it very hard to synthesize proteins. It certainly isn't a situation that would get your body to respond by conserving proteins and building more muscle with it.
I'm not saying it's 100% wrong, just that it's unlikely and I'm not really sure it's at all right. Honestly, I've had some similar theories between fat and carb conversion that would lead to the idea that burning fats is actually not a very good way to lose bodyfat, and thus things like keto diets are counter-productive.
I'm trying to see some way that Menno's ideas can work, and they might be there - but he can't explain them sufficiently and his math is a mess, and he is flat out wrong about some things.
I do take issue with his "proof" being a sole example of a person who had a dexa scan and he is claiming the person gained 6+ pounds of muscle in a month. Well, if they did, we can suspect steroids right off the bat. But what I really suspect is that he is being dishonest about the fact that dexa scans don't measure muscle mass - they measure bone mass, fat mass and everything else (water, proteins) is lumped into the same category. So, if I get a dexa scan right now and then drink a gallon or two of water without peeing, and don't eat anything before I go back tomorrow and get another scan, I will skew my fat mass down slightly (more water throws off the scan) and dramatically increase my "muscle mass" in his eyes because the water weight is not differentiated from muscle weight.