Why it is a question
On Monte Carlo samples of the 2D Ising model and the noisy 2D dimer model, the block coarse-graining trained to maximize mutual information with the environment beyond a buffer recovered the variables renormalization already knew: Kadanoff block spins and the dimers’ low-momentum electric fields. Iterated on the Ising model, it gave the flow and ν ≈ 1.0 ± 0.15 (THEORY-194, from LIT-873 read in NOTE-674). The sequels extend the evidence to the interacting dimer model across a BKT transition (LIT-878), again with operators known from field theory. They also prove that a coarse-graining keeping all of a block’s information about its environment adds no range to the Hamiltonian, while the optimised coarse-grainings fall short of that condition (LIT-881, THEORY-198). Near the first information-bottleneck transition the optimal encoder of a slab follows the leading transfer-matrix mode (LIT-885, THEORY-205), which is a different geometry from a block in a buffer shell.
So whether the objective and renormalization agree where renormalization’s variables are unknown is open, and so is whether the maximizer yields short-ranged couplings in two or more dimensions. Until 2026-10-20 this was THEORY-194’s promote_when.
What would answer it
Yes, empirically: a system in two or more dimensions whose relevant variables were not known beforehand, where the maximizer is computed and iterated and gives critical exponents that agree with an independent method (high-precision Monte Carlo, series or conformal bootstrap) within stated errors, while a coarse-graining fitted to the data distribution on the same samples does not. Yes, formally: a proof, for a class of local Hamiltonians in two or more dimensions, that the maximizer yields an effective Hamiltonian whose couplings decay with distance.
No: a system where the maximizer’s iterated flow gives exponents that disagree with an independent method, or a local Hamiltonian in two or more dimensions where the maximizer produces long-ranged couplings.
What would not answer it: more rediscoveries of block spins or of other variables already known, since the account was shown on exactly those.