Why it is a question
THEORY-585 now states a theorem about Döring and Isham’s scheme (LIT-310): for finite-dimensional systems, the pull-back of a constituent’s quantity agrees with the composite’s own quantity at a context exactly when every projection in the context with proper support on the first factor has the form P⊗1, so it fails for every non-scalar quantity, and no operator on the composite realises the pull-back. That is a result about these rules: the functor W ↦ V_W, outer daseinisation, and a context category augmented with the trivial algebra. The authors call the rules “partly experimental”, and the trivial-system rule conflicts with the series’ own definition of contexts. Whether some other choice repairs the composite is open, and so is the infinite-dimensional case.
What would answer it
A variant of the rules or of the context category, for instance a different pull-back functor or a different daseinisation, under which the translation condition holds for tensor products would answer it yes, and would bear on what THEORY-585’s failure is a failure of. A proof that no pull-back of the scheme’s general kind satisfies the condition for H₁⊗H₂ would answer it no. A further check on disjoint sums cannot settle it: they already compose exactly.