This is Budo. One of the primary elements in Budo is personal honor. You make a promise not to teach without permission and you keep it. Even if it is no longer convenient for you. That was one of the terms of being accepted as a student. That promise doesn't go away if you leave the dojo of your own free will. It doesn't go away if you are thrown out of the dojo. It doesn't go away unless the person you made the promise to releases you from that promise.

Of course, the training you received doesn't simply disappear. If you keep training, perhaps, after many years, what you are doing does morph into something completely yours. At that point it might make sense to give it a new name and be the Founder of some new art. But i this case we are talking about a 4th Kyu. He has not devoted years and years to his own development of this style after leaving his Teacher. He simply gave it a new name and went down the street and started trying to teach. This is in direct violation of his promise not to do so. It is unethical and dishonorable. It's no different than taking someone else's intellectual property as ones own.

In all the cases of folks violating their promises by teaching, there are always people who do not think there is anything worn with it. There are folks who think anything one can download should be free. That any DVD can be copied and freely distributed. No one should be told they can't do whatever they want to do. After all, it's a free country and who is anyone to tell anyone else what to do?

But as far as I am concerned, this stuff matters. You promised you wouldn't so you don't. This is the overriding issue, integrity. Then you can get to the issue of 4th kyus "founding" their own styles. Next thing you see is membership in the World Sokeship Council with a bunch of fake sokes certifying other fake sokes, and blah, blah, blah.