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LIT-885: Relevance in the Renormalization Group and in Information Theory

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StatusActive — read 2026-10-09 (NOTE-690); worth reading as the theorem the real-space mutual-information programme (LIT-873, LIT-881, LIT-878) cites for its claim that the optimal coarse-graining picks out the most relevant operators. What it proves is narrower than that claim: for a short-range lattice model on a cylinder, with the block and the environment taken as whole slabs of the cylinder separated by a buffer, the information-bottleneck encoder just past its first transition depends on the block only through the boundary weak value of the leading transfer-matrix eigenvector, and the transition sits at β_c = (λ0/λ1)^(2L_B); for large circumference that eigenvector is the lowest-dimension primary. The planar block-in-a-shell geometry the algorithms use, the fixed-alphabet β → ∞ limit that is RSMI, and the regime beyond the first transition are not covered. A second result: past the first transition, with a large enough alphabet and no fine-tuning, the optimal encoder carries a representation of the data's symmetry, and a two-symbol example with Z4 symmetry breaks it.
Tagsnatural-sciences, information-theory, representation-learning
ExtendsLIT-873 — Mutual information, neural networks and the renormalization group
Cited as “paper” byNOTE-690 — Relevance in the Renormalization Group and in Information Theory
Cited as “source” byTHEORY-194 — On Monte Carlo samples of the 2D Ising model and of the 2D dimer model with added noise spins, a block coarse-graining trained to maximize mutual information with the system beyond a buffer recovered the variables renormalization already uses (Kadanoff block spins; the dimers' low-momentum electric fields, with no weight on the noise), and iterated on 128 × 128 Ising samples gave the flow away from the critical point, locating it to about 1% and giving ν ≈ 1.0 ± 0.15, while a restricted Boltzmann machine trained by contrastive divergence on the noisy dimer data coupled to the noise and to columnar patterns instead, THEORY-205 — For a short-range lattice model on a cylinder with a symmetric transfer matrix whose leading excited eigenvalue is non-degenerate, with block and environment as slabs L_B transfer steps apart, the uninformative information-bottleneck encoder first loses stability at β = (λ0/λ1)^(2L_B), in the direction of the boundary weak value of the leading excited eigenvector, which by the cited conformal finite-size relation is the lowest-dimension primary at large circumference; the uninformative encoder can stop being optimal at a smaller β, and nothing is shown for blocks inside a buffer shell or for the fixed-alphabet RSMI optimum

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Amit Gordon, Aditya Banerjee, Maciej Koch-Janusz and Zohar Ringel (2020), Physical Review Letters 126(24):240601 (2021) — ARXIV-2012.01447, DOI-10.1103/PhysRevLett.126.240601

Key takeaways

  • The theorem, in its geometry. A short-range lattice model on an infinite cylinder of circumference L; the block V and the environment E are whole slabs of the cylinder, separated along its axis by a buffer of L_B slices. Through the transfer matrix T, the joint law reduces to P(v|e) = P(v)[1 + ε r_v r_e] up to terms of order (λ2/λ0)^(L_B), with ε = (λ1/λ0)^(L_B) and r_v = ⟨1|∂V_R⟩/⟨0|∂V_R⟩ the weak value of the leading excited eigenvector on V’s boundary slice facing E. The information-bottleneck (IB) encoder then satisfies P(h|v) ∝ P(h) exp(β ε² r_v ⟨r_v⟩_h): it sees V only through r_v.
  • The first transition, in closed form. The trivial encoder loses stability at β_c⁻¹ = ε², and just above, for a binary H, P(h|v) = e^(h m r_v)/2cosh(m r_v) with m growing as √(β − β_c). As L → ∞ with L_B ≫ L, ε → exp(−2πΔ₁L_B/L) and r_v belongs to the primary of lowest scaling dimension Δ₁: IB-relevant is RG-relevant. The identification with a CFT is used only to interpret the transfer-matrix result, which the authors stress holds at any L once L_B ≫ L.
  • One numerical check. Critical 2D Ising on a cylinder of circumference 3, |V| = 2 × 3, |E| = 1 × 3, L_B = 9, the joint law computed exactly: an iterative IB solver finds the transition between 146.340 and 146.350 against a predicted 146.34458, and the encoder follows the boundary magnetization. The prediction is reproduced exactly in NOTE-690.
  • Symmetry. If P(e, v) is invariant under a group S acting by permutation, then for β below the second transition, with an alphabet large enough that one more symbol does not help, a second-order first transition and no fine-tuning, the optimal encoder carries a permutation representation of S. With V = E = Z4 and |H| = 2, at β → ∞, the optimum breaks the Z4 symmetry (checked by hand in NOTE-690). A procedure for reading an unknown symmetry off a solved encoder is sketched.
  • What is asserted, not shown. That RSMI (LIT-873) is the β → ∞ limit of IB at fixed |H|, so that the theorem underwrites it; that the whole IB curve of the continuum Gaussian theory is computable (cited to work in preparation); and that the symmetry result holds at all β (stated as a conjecture).

Standing in the record

Filed on 2026-10-09 at the owner’s request, as one of the works cited by the batch on hierarchy and hyperbolic geometry (nucleation#113, LIT-865 to LIT-877) and its cited works (nucleation#114) that neither record held. It is cited by Gökmen, Ringel, Huber and Koch-Janusz, Statistical Physics through the Lens of Real-Space Mutual Information (LIT-878), which takes from it that the RSMI-optimal filters are the most relevant operators and calls the result “proven in part”. Read on its own merits (NOTE-690), after LIT-878 and NOTE-681, LIT-873 and LIT-881.

Does the proof hold? In its own setting, yes, and more exactly than the paper says: in the slab geometry the factorization through the boundary slices is exact, so β_c⁻¹ = ε² and the critical direction r_v hold with no corrections, which a transfer-matrix computation reproduces to every printed digit. What it assumes is that setting: a real symmetric transfer matrix, block and environment spanning the whole circumference, a non-degenerate leading eigenvalue, the neighbourhood of the first IB transition, and, for the scaling-dimension reading, L → ∞ with L_B/L → ∞. It does not reach the planar block surrounded by a buffer shell that LIT-873 and LIT-878 use, nor RSMI’s fixed-alphabet, β → ∞ optimum, which the paper relates to IB by assertion. So LIT-878’s reliance is on a theorem for a neighbouring geometry and regime; “proven in part” is fair. NOTE-690 gives a leading-order argument, the reader’s and not the paper’s, that closes the RSMI gap in the slab geometry, and says why the planar case is still open.

It is filed with THEORY-205, the formal counterpart of the empirical account THEORY-194 and a different statement from THEORY-198’s full-capture factorization. It bears on QUESTION-025 only by analogy, and on CLAIM-092 only as an instance of what an information bottleneck keeps being characterized spectrally: see NOTE-690.

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