Foundation of Problem Solving Some starting considerations, given where we are...
The foundation of problem solving is a name that - if adequate - will somehow pick out what all problem solving cases have in common, in terms of basic elemental constituency. The scope of the minimum criterion must be as broad as possible, ranging from the most simple, rudimentary, and/or basic examples of problem solving through and perhaps 'beyond' the most complex cases known.
We are actually problem solving here and now, in this very thread. We are looking to acquire knowledge of what all problem solving has in common, how it evolves and/or grows in it's complexity, etc. So, if there is a single foundation underlying all of the different complexity levels, then it must be amenable to each and every one. If what we propose as the foundation of problem solving cannot adequately account for all known cases, then it is not the foundation. To quite the contrary, it would be utterly inadequate; otherwise sorely lacking, in it's explanatory power. If there is a foundation for problem solving, then all examples thereof must somehow, and in some way or other, be commensurate with and/or otherwise amenable to their own foundation.
In light of these considerations, there needs to be yet another; a standard to bear. All candidates worthy of subsequent time and consideration - whatever they may be - must be amenable to evolutionary terms. This is already supported by and in large by the simple things we already know. All human knowledge is accrued. It begins simply and grows in it's complexity.
Any and all foundational accounts are and must be put in bare minimalist terms, for there is no other way to be able to 'connect' all the cases ranging from what you've called "unconscious" through extremely complex metacognitive endeavors, such as what we're actually doing here and now. In this thread, we are thinking about our own thought and belief, with a particular 'kind' of them in direct view.
A foundation is never equivalent to what's built upon it. Sometimes, the foundation consists of entirely different elemental constituents(think of a building). Other times, however, the foundation consists of the very same things albeit in much simpler 'form' as compared to the more complex things emerging and/or growing in complexity from that simple basic elementary 'form'.
The foundation of problem solving is one such thing.