Yes, but the only part of the fat that transfers to the water is the hydrogen, which weighs just slightly more than nothing (perhaps TFA covers this... I didn't read it...)
> ... the only part of the fat ...
It's only Hydrogen... But it still needs to go somewhere. Oxygen comes in to carry away the Hydrogen as water--you can't urinate Hydrogen.
You can urinate other things than water though, some others of which contain hydrogen.
Sure, my point is just that the carbon outweighs the hydrogen ~6:1.