I think we should also rule out positions with either king in double check, since almost all such positions are impossible. My guess at the most populous combination over all chess positions is for each side to have one pawn, three rooks, three knights, three bishops, and three queens, after losing two pawns each to pawn captures to make the remaining pawns promotable. As I mentioned in the message to Rich last week, the possible positions are composed of a near-unanimity of wildly implausible positions. Apologies for a slight detour, but I'd like to comment on this. A more efficient theory is one which can make solid predictions with much smaller expenditure of effort than traversing the entire move tree to its uttermost ends.
Theories appear to be a singular asset of cultures, having direct bearing on evolutionary viability.
Theories are valued by their completeness, effectiveness and cost of application. So far, absent the move tree, we construct only provisional theories of chess. I don't want to start some philosophical thread here, only to address the above, rather narrow, vision of what this effort can contribute. I'd like to comment on this. A complete theory of chess is almost surely far beyond human comprehension.