Sorry to butt in.
Richard, I understand full well the vagueness that comes together with the Japanese term of "breathing," but I think what Mike is gunning for is more a "what happens" when you breathe.
How does breathing in or breathing out contribute to power, how does it aid the body, or what effects do certain types of breathing have on the body.
The answer, at least from my own experience, isn't some vague answer that deals with relaxation, "matching", etc etc
There's a more practical effect on the body, mainly in terms of conditioning. Course, you have to have a certain amount of conditioning to feel it...if you don't then (and that "you" is not aimed at "you," I meant it generally) you won't be able to feel it.
Case in point, the one demo that you gave with the student gripping the end of the sword as the teacher raises and lowers it, while the student is unable to punch/kick, seems pretty easy, given someone with enough conditioning in the body (not of muscle tho).
Launching the guy on the downswing should be fairly easy for anyone conditioned enough, and could be indicative of someone that's trained to use the breath to store and release power.
Well, I have been able to do that sword thing myself, but I did not do any particular "Breath Method" in order to accomplish it. I was not able to do it with the jo. The distance between uke and I was just too great (for my level) in regards to the jo. I would have to try this technique again (with breath limitation - ie specifically not breathing through the technique) in order to find out if breath had anything to do with it or not. Maybe I am naturally using my breath correctly when I successfully do this technique and thusly my few successes were due to having developed some breath skill or lucky breathing. I will play with over a period of time and see what comes of it.
Do you feel you have developed enough breath conditioning to do this sword thing or even the jo thing? If so, I would love to hear how your experiments with it go when you try it out. Try out the other one too (maybe your theory will make it work for you).
Your breath theory reminds me of something I learned from Ushiro Sensei. It seems to me that he uses breath as a way to create even expansion in his body (like a sphere), which makes him very strong without the use of external muscular strength. I messed with that breathing for a while and found that it was useful for a variety of things (being able to absorb blows) and preventing people from taking your balance, etc. Most people when they breath in expand their chest which causes the upper spine to change position and bend slightly backwards on the intake and then go forward again on the exhale. This is a loss of structural integrity. In normal breathing this is happening to everyone but on such a small level that they don't notice it. If you ask someone to take in a large breath and observe how their body moves when they do it, it will become obvious. If you point this out to them, then they will easily perceive it in themselves. Of course there are other biological aspects to proper breathing, like endurance, carbon dioxide expulsion which clarifies thinking and heightens awareness increases relaxation, etc.
If I recall correctly, Kuroda Sensei could do all his stuff while not breathing. I seem to recall a discussion about "breath power" in the dojo. As I don't train there now, I can't be sure on that account though.