Materialism and idealism mean, respectively, that matter exists even though there isn’t an observer, and that it stops existing if there isn’t an observer.

Regarding the field, why wouldn’t it be materialist? It is a force that exists in the same dialectical materialist world as matter. Dialectical materialism, which is the kind of materialism Engels talked about, is a combination of what I’ve described above and a rejection of the metaphysical idea that everything remains the same, in dialectics everything changes. That’s why he mentions the “matter in motion”.

I don’t understand some of the mathematical jargon you use but I feel you are a bit confused with the diamat interpretation of matter. I feel like if anything, quantum mechanics prove exactly the point of diamat, because it is a contradiction, which is exactly the point of it. Something that both exists and that it doesn’t.