A supernatural thing is a thing not governed by law. I expect most supernaturalists will immediately object that I have omitted the word "natural" before the word law, arguing that supernatural causes may be governed by unnatural laws.
Here, I'm going to dispense with this confusion. There's really no natural/non-natural distinction when it comes to law. I can't think of any non-natural laws. A thing (or aspect of a thing) is natural if and only if that thing is governed by law.
Mathematics is lawful and non-physical, yet we don't consider mathematics to be supernatural.
Electromagnetic phenomena like lightning were once considered unnatural phenomena, but electromagnetism became natural when it was found to obey laws. So, the whole idea of "supernatural law" is quite ridiculous. A supernatural law would simply be reclassified as a natural law in the same way lightning is now interpreted as a natural phenomenon.
This means that the space of supernatural stuff contains nothing governed by laws of any kind. No matter what we observe about a supernatural system or precursor, we cannot say anything about its future state.
Have I proven that there are no supernatural (uncaused) events? Not at all. There may well be such events. It's just that, if we knew that an event were supernatural, its supernatural nature would be all we could ever know about it. Of course, we can't actually know that an event is supernatural. At best you might show that you cannot find a law that would account for the event. Hence, the assertion that an event is supernatural is the assertion that not only do we not know the cause of an event, but we cannot know anything about its cause, not even in principle. It's ignorance piled on top of ignorance.