This examination of reality is running like an exciting detective novel. The research uses a hierarchy of spaces that show varying degrees of complexity. Finding more complicated spaces is not a big problem. This turns out to be quite simple. The move to a more complicated form is always accompanied by several significant limitations and understanding where these limitations come from is a much bigger problem. Today's mathematics is only just able to explain the limitations of the Hilbert space. This no longer applies to the system of Hilbert spaces, which in this document is called the Hilbert repository. Current math cannot explain the restrictions and the extra features of the Hilbert repository. This fact will particularly interest those who are curious about the structure and behavior of physical reality. The approach is quite different from the usual path and provides other insights. The usual way tries to deduce new insights from what we know about classical physics. The story makes it clear that only mathematics cannot provide a complete picture of reality, while experiments alone also cannot expose physical reality. The combination of mathematics and experimentation produces the best results.