Why it is a question
THEORY-198, narrowed on 2026-10-19, holds that a block coarse-graining keeping all of the block’s information about the environment beyond a buffer makes the coarse measure factorize across the block. This is proved in 1D, and in D dimensions when capture by a hyperplane of blocks amounts to capture by each block and no existing couplings cancel. So the renormalized Hamiltonian gains no range. Until that date, THEORY-198’s promote_when asked about the case short of full capture. At the owner’s direction, from the promotion audit’s batch 019 (ADR-047), the THEORY was narrowed and promoted on the proved case, and the condition moved here unchanged.
It is open because full capture is not what optimization reaches. With the number and type of coarse variables fixed in advance, the maximizer usually falls short: decimation of two-spin Ising blocks keeps about half of I(V : E) (0.505 at K = 0.1 with a one-site buffer, NOTE-682). Whether the information lost bounds the couplings generated is shown only by perturbative calculation on 1D chains (LIT-881), and only locally monotonically. It refines QUESTION-065’s question of whether locality can supply the target a sufficiency-based boundary is relative to.
What would answer it
The condition formerly on THEORY-198, unchanged. It asks for a proof that the coarse-graining maximizing mutual information with the environment beyond a buffer, under a fixed number and type of coarse variables and so short of full capture, yields effective couplings whose range or strength beyond nearest neighbours is bounded by the information lost, for a class of local Hamiltonians in two or more dimensions. Or it asks for a check, on a 2D lattice model with its blocks, that the per-block capture assumption of the D-dimensional proof holds or fails. More 1D cases cannot settle it. Nor can examples in which the renormalized Hamiltonian is short-ranged for reasons of the model: decimation of the Ising chain is exact whatever information it keeps.