I believe I have characterised it in the description I gave of it.
Your characterisation, such as "non spatio temporal", is inept as NS has pointed out - how can something be said to exist outwith time or space: how could you identify it in the first place?
In terms of knowledge. That I know about it doesn't affect that things existence or otherwise does it?
But how do you 'know' about it, Vlad: if the existence of this "non spatio termporal" thing is indeed known to you then what items of knowledge do you have about it, epistemology-wise?
For example, on re-reading this particular point of yours, you seem to be saying that there is a difference between what you think you 'know' and the "existence or otherwise" of the thing you think you know about, and this seems to open up the possibility that you think you have knowledge of something that might not exist - and that makes no sense.
There is a lot of talk in the scientificcommunity that Maths is the basis of reality and Maths would certainly fit the description I gave.
But Maths is axiomatic, Vlad, and not a 'thing' that can be said to exist "non spatio temporally" - I think you are grasping at things thoughtlessly.
In view of this i'm not sure the question ''why would you think that there was more than the material'' is particularly an intelligent one.
It is a perfectly intelligent question to at least ask: remember the risk of those pesky 'unknown unknowns' cannot just be dismissed, but it wouldn't be intelligent to answer that there could be more than the natural, as opposed to "material", in the absence of a method suited to the non-natural, since without such a method 'non natural' is a meaningless term.