Source
Gordon, Banerjee, Koch-Janusz and Ringel (2020; Phys. Rev. Lett. 126, 240601, 2021), LIT-885: main text Eqs. 3–5 and Fig. 3; Appendix B (Eqs. B1–B9), Appendix C (Eqs. C1–C11), Appendix F; as read in NOTE-690.
What was actually shown
The setting. An infinite cylinder of circumference L carrying a short-range lattice model with transfer matrix T. The block V and the environment E are runs of whole slices, separated along the axis by L_B slices. The relevance variable of the information bottleneck (IB) is E.
The derivation. Writing all distributions through powers of T and its eigendecomposition gives P(v|e) = P(v)[1 + ε r_v r_e] up to terms in (λ2/λ0)^(L_B), with ε = (λ1/λ0)^(L_B) and r_v the ratio of the leading excited to the ground eigenvector on V’s boundary slice. The IB equations then reduce to P(h|v) ∝ P(h) exp(β ε² r_v ⟨r_v⟩_h). Linearizing around the uniform encoder puts the first transition at β_c⁻¹ = ε², with the encoder’s first non-trivial direction r_v; a third-order expansion gives the binary encoder above it in closed form. As L → ∞, λ_i/λ0 → exp(−2π∆_i/L), so r_v belongs to the lowest-dimension primary.
What could have come out otherwise. On a critical Ising cylinder of circumference 3, an iterative IB solver given the exact joint law could have found its transition elsewhere, or an encoder not following the boundary spins. It found 146.340 < β_c < 146.350 against a predicted 146.34458, and the predicted encoder. NOTE-690 reproduces the prediction exactly from the singular values of the normalized block–environment channel, and notes that in this geometry the reduction is exact rather than leading-order: the Markov property along the cylinder makes P(v, e) a sum of rank-one terms with weights (λ_i/λ0)^(L_B), so β_c = (λ0/λ1)^(2L_B) with no correction.
What this does not say
- Not that this is where the uninformative encoder stops being optimal. The theorem is a linear-stability result. Below the bifurcation an encoder far from uninformative can already beat it, which happens on a skewed 3-state chain (dated note below). The two thresholds agree in the Ising check, and when they agree in general is QUESTION-185.
- Not that IB and RG relevance are equivalent in general, as the paper’s abstract puts it. The theorem concerns the first IB feature in one geometry. Later transitions are not derived.
- Not that RSMI filters are the most relevant operators. RSMI (LIT-873, LIT-878) maximizes I(H;E) at a fixed alphabet, which the paper calls the β → ∞ limit of IB without analysing it, and its blocks are planar squares inside a buffer shell, not slabs. NOTE-690 sketches a leading-order argument that closes the first gap in the slab geometry; the second is open.
- Not dependent on criticality, except for the reading. At any temperature the first IB feature is the leading excited transfer-matrix mode; only calling it a scaling operator with dimension ∆₁ needs a CFT.
- Not a result about short-ranged effective Hamiltonians. That is THEORY-198’s subject (full capture keeps the Hamiltonian short-ranged); this account says which feature a minimal capture takes, and neither implies the other.
- Not covering degenerate leading operators. With several operators of the same lowest dimension the IB fixes a subspace; the paper’s symmetry argument says the encoder then carries a representation of the symmetry when the alphabet is large enough, and gives a Z4 example where a two-symbol alphabet breaks it.
- Not an instruction for machine learning. It says what an information-bottleneck optimum is in a lattice model, not how to train anything.
Connections
It is the formal counterpart of THEORY-194, which states on numerical
evidence that the coarse-graining maximizing information with the far
environment is the RG-relevant one: it proves the operator half of that
claim in a neighbouring geometry and regime, and does not meet
THEORY-194’s promote_when, which asks for unknown variables found or a
proof about effective Hamiltonians in two or more dimensions. It sits
beside THEORY-198 (LIT-881) as the second theorem of the same programme.
It is a worked analogue for THEORY-161 and CLAIM-092, which pose meaning
and translation as information bottlenecks: here what the bottleneck
keeps first is computed exactly, and it is the leading singular direction
of the channel. That is an analogy, not evidence for either.
Note of 2026-10-19: the derivation, re-derived
The setting: T symmetric, with eigenvalues λ0 > |λ1| > |λ2| ≥ ⋯ and orthonormal eigenvectors ψ_i, where ψ0 > 0 by Perron–Frobenius. Write r_i = ψ_i/ψ0, and let n be the number of transfer steps between V’s boundary slice s and E’s boundary slice t.
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The joint law. A slice’s marginal is π = ψ0², and the n-step law of t given s is π(t) Σ_i (λ_i/λ0)^n r_i(s) r_i(t). Along the cylinder E is independent of V given s, and V of E given t. So P(v, e) = P(v)P(e)[1 + Σ_{i≥1} ε_i r_i(s(v)) r_i(t(e))], with ε_i = (λ_i/λ0)
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An exact singular-value decomposition. Since Σ_s ψ_i ψ_j = δ_ij, the r_i∘s are orthonormal and mean-zero under P(v), and so are the r_i∘t under P(e). So step 1 is an exact singular-value decomposition of P(v,e)/√(P(v)P(e)), with singular values 1 and |ε_i|. Nothing is dropped, as NOTE-690 observed.
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The second variation. Perturb the uninformative encoder, p(h|v) = p(h)(1 + δ g_h(v)) with each g_h mean-zero. To second order in δ, I(H;V) − β I(H;E) = (δ²/2) Σ_h p(h)(‖g_h‖² − β‖K g_h‖²). Here K takes a function of V to its conditional expectation given E. Since K(r_i∘s) = ε_i r_i∘t, the largest ‖Kg‖²/‖g‖² over mean-zero g is ε_1², reached only at g = r_1∘s. So the uninformative encoder first loses stability at β = ε_1^(−2) = (λ0/λ1)^(2n), in the direction r_1(s).
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The Ising check. For critical 2D Ising on a cylinder of circumference 3, with n = 9, the eigenvalues are λ0 = 17.87705 and λ1 = 13.55181. Both (λ0/λ1)^18 and the inverse squared singular value of the exact channel are 146.34458, the value LIT-885 predicts. So L_B counts transfer steps.
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Global optimality is a different threshold. The uninformative encoder is optimal exactly when I(U;V) ≥ β I(U;E) for every encoder U. That fails once β > 1/s*, where s* = sup I(U;E)/I(U;V). Since s* ≥ ρ_m² = ε_1², this threshold is never later than the bifurcation.
- Ising: s*/ρ_m² = 0.999999 numerically, so the two coincide.
- A skewed 3-state chain: take T = [[1, 0.9, 0.05], [0.9, 1, 0.05], [0.05, 0.05, 0.3]] with n = 3. The bifurcation is at (λ0/λ1)^6 ≈ 69,391. A binary encoder that flags the rare state with probability 10⁻⁴ already gives I(U;V) − 5000 I(U;E) < 0, and s* ≈ 27 ρ_m².
So the theorem locates the bifurcation, and not in general the first transition of the global optimum. The encoder that wins on the 3-state chain still separates states by the sign of r_1. Whether the global optimum’s first feature is always the leading excited mode is part of QUESTION-185.