Source
Lenggenhager, Gökmen, Ringel, Huber and Koch-Janusz (2018; Phys. Rev. X 2020), LIT-881: Sec. III and Appendix B (Lemma, Propositions 1 and 2, Eqs. B1–B22), Secs. IV and VI with Appendix D (Figs. 4, 5, 9–12); as read in NOTE-682.
What was actually shown
The proof. Blocks are taken large enough that only neighbouring blocks interact. Fix a block (in D dimensions, a hyperplane of blocks), call its neighbours the buffer and the rest the left and right environments. If the coarse variable keeps all the block’s information about the environment, the block carries no further information about the environment given the coarse variable; locality makes the two environments independent given the block; together they make the two environments independent given the coarse variable. The coarse measure then factorizes across that variable once the coarse buffer is summed out, so no effective term couples the two sides. Since the block was arbitrary, the range does not grow. In 1D with a product distribution of nearest-neighbour disorder, the same rule stays optimal under any change of disorder confined to one side, so the renormalized disorder distribution carries no correlations across the block.
This could have failed: full capture might have left the environments correlated through the buffer, and in D dimensions it might require coarse variables shared across blocks. The first is excluded by the proof; the second is excluded by assumption.
The numerics. For every two-spin rule (λ₁, λ₂) on the 1D Ising chain and on the dilute random Ising chain, the information kept is computed exactly and the renormalized couplings perturbatively. Next-nearest neighbour and four-spin couplings, and correlations between neighbouring renormalized couplings, vanish where the information is greatest, at decimation. Majority rule, which keeps least, generates them.
What this does not say
- Not that the maximizer gets full capture. With the number and type of coarse variables fixed, it usually cannot. In the paper’s worked case decimation keeps about half of I(V : E) (0.505 at K = 0.1 with a one-site buffer, by the reader’s calculation), yet its Hamiltonian is exactly nearest-neighbour; that exactness is a property of the 1D chain, not a consequence of the theorem.
- Not that complexity falls monotonically with information. The authors find it monotonic only locally, with accidental zeros away from the maximum; what is kept is that the global maximum is a global minimum of the measured couplings.
- Not a proof for two or more dimensions without conditions. The D-dimensional step assumes that capture for a hyperplane amounts to capture by each block separately, called reasonable, not shown, and excludes fine-tuned cancellations of existing couplings. The disorder result is proved in 1D only.
- Not that the block, buffer and environment are found. They are a spatial partition fixed before anything is computed, and locality is stated with respect to it.
- Not an instruction for machine learning. No network is trained; the RBM is only a family of rules.
Connections
It underpins THEORY-194, which records that the information-maximizing
coarse-graining matched the known RG variables of the Ising and dimer
models: this is the argument for why such a coarse-graining should keep
the effective Hamiltonian short-ranged. It does not meet THEORY-194’s
promote_when, which asks for a statement about the maximizer, not the
perfect case. The Lemma is a screening statement of the kind THEORY-073
examines for Markov blankets, with the screening region likewise placed
in advance; that parallel is the reader’s. It does not bear on
QUESTION-025.
Note of 2026-10-19: the Lemma, re-derived, and the narrowing
Written for the promotion audit’s batch 019 (ADR-047). The re-derivation is in record/promotion-audit/decisions/THEORY-198.yaml and is restated here.
Order the 1D chain E_L, B_L, V, B_R, E_R, with V one block, B_L and B_R the buffer blocks beside it, and E_L, E_R the rest. H is V’s coarse variable, drawn from P(H | V), so H is independent of everything else given V.
- Full capture screens. By the chain rule, I(V, H : E) = I(V : E) + I(H : E | V) = I(V : E), and also I(V, H : E) = I(H : E) + I(V : E | H). So full capture, I(H : E) = I(V : E), gives I(V : E | H) = 0.
- Locality. Only neighbouring blocks interact, so the Gibbs measure is Markov along the chain and V separates the two sides: E_L ⊥ E_R | V.
- Combined. For any v with P(v | h) > 0, step 1 gives P(e_L, e_R | h) = P(e_L, e_R | v), which step 2 splits into P(e_L | v) P(e_R | v). Step 1 again, marginalised, turns each factor into P(e_L | h) and P(e_R | h). So E_L ⊥ E_R | H.
- To the coarse lattice. The coarse variables of E_L’s blocks are drawn from E_L’s spins alone, and likewise for E_R, so the independence passes to them. With the coarse buffer summed out, P(X′{≤−2}, X′{≥2} | X′0) = P(X′{≤−2} | X′0) P(X′{≥2} | X′_0).
No term of that marginal measure couples the two sides across X′_0. Reading this as “the renormalized Hamiltonian gains no range” needs the source’s exclusion of fine-tuning: summing out the coarse buffer must not exactly cancel a longer coupling. The title now says so. The D-dimensional proof runs the same steps on hyperplanes of blocks. It needs full capture for the hyperplane to amount to full capture by each block, which the source calls reasonable and does not show, and the title now states that too.
The promote_when asked what happens short of full capture, which this account disclaims. It moves, unchanged, to QUESTION-154.