Why it is a question
THEORY-202 holds the general case. A joint Hamiltonian can match the partition function with the wrong marginal. A hidden system that does not interact with the visible one meets the trace condition exactly, with a coarse Hamiltonian chosen freely. That second construction needs a general Boltzmann machine. Kadanoff’s schemes fix the kernel’s form, so that each hidden spin couples to a block, which excludes it. Whether exactness inside those families keeps the relevant operators, or still leaves the coarse variables free, is not settled by either source. Until 2026-10-20 it was THEORY-202’s promote_when.
What would answer it
- A treatment of variational renormalization, read and checked, that states a further condition under which a coarse-graining counts as a renormalization (a fixed kernel form, a short-range condition on the coarse Hamiltonian, or the preservation of named observables) and shows that the trace condition does not imply it inside Kadanoff’s families.
- A proof that, inside those families, the trace condition does force the coarse Hamiltonian to keep the relevant operators.
What cannot settle it: more examples of coarse-grainings that preserve Z or satisfy the trace condition.