Contribution
It ties Hoel’s effective information to a spectral quantity and uses the spectrum to define causal emergence without naming a coarse-graining. A TPM is “dynamically reversible” if it is invertible with a stochastic inverse, which holds exactly for permutation matrices; Γ_α, the Schatten α-quasi-norm to the power α, measures how near a TPM is to one. Emergence is then the gain in Γ per retained dimension when small singular values are cut. A coarse-graining by SVD projection and k-means is offered for comparison with EI maximization.
Key insight
EI is high when rows of the TPM are distinct one-hot vectors, and so is Σσᵢ: both peak, at N and log N, exactly at permutations. Near-zero singular values mark redundant “information pathways” (rows that are nearly linear combinations of others). Discarding them raises the average singular value, and that rise is a coarse-graining-free stand-in for what a good coarse-graining would gain in EI.
Assumptions
- A known discrete Markov chain, TPM P, N states; EI with uniform interventions as in Hoel et al. 2013.
- α ∈ (0, 2) in Definition 2; α = 1 (the nuclear norm) in all experiments. Proposition 3 is stated for α ∈ (0, 1) and proved for (0, 2).
- A threshold ε for vague emergence, chosen “according to the relatively clear cut-offs in the spectrum”.
- Irreducible, recurrent P with a stationary distribution, for the coarse-graining method of §4.
Key results
- Proposition 1. P is dynamically reversible iff it is a permutation matrix. Lemma 3: dynamical reversibility implies time reversibility.
- Proposition 2 and Lemma 9. Γ_α ≤ N with equality iff P is a permutation; Γ_α ≥ 1 with equality iff every row is uniform. Lemma 11: Γ_α ≥ ‖P‖_F
- Theorem 1. log Γ_α − log N ≤ EI ≤ (2/α) log Γ_α. The tighter EI ≤ log Γ_α is seen numerically and left unproved; EI ∼ log Γ_α is conjectured from simulated matrices.
- Definitions 3–4. Clear CE: rank r < N, degree ∆Γ_α = Γ_α(1/r − 1/N). Vague CE at ε: ∆Γ_α(ε) = Σ_{i≤r_ε} σᵢ^α / r_ε − Σ_i σᵢ^α / N, with r_ε = max{i : σᵢ > ε}. Proposition 4: 0 ≤ ∆Γ_α(ε) ≤ N − 1.
- Examples. Hoel’s 4-state Boolean network: 4 non-zero singular values, ∆Γ = 0.75; with Gaussian noise (std 0.03), ε = 0.2 gives ∆Γ = 0.69. A 6-node network (64 states): 8 non-zero values, ∆Γ = 2.23, Γ = 20.39, γ = 0.32, reduced to γ = 1.0. A 100-node, 5-block SBM: ε = 0.3, r_ε = 5, ∆Γ = 0.56. Local TPMs of elementary CA rule 40 show clear CE at some cells.
- Coarse-graining (§4). Project rows on the top r_ε right singular vectors, cluster by k-means into r groups, lump by stationary flow (Klein and Hoel’s method). On Hoel’s examples it gives the same or a higher-EI macro chain (E.1).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Γ_α is maximal exactly at permutation matrices | strong | Proposition 2, proof checked |
| C2 | EI is bounded above and below by affine functions of log Γ_α | strong | Theorem 1, proof checked |
| C3 | EI ∼ log Γ_α | weak | simulated softened permutations, softened degenerate and random TPMs |
| C4 | CE can be quantified independently of any coarse-graining | moderate for ε = 0; weak for ε > 0 | Definitions 3–4; the ε > 0 measure moves the choice into ε |
| C5 | The SVD and EI-maximization frameworks are equivalent | weak | agreement on Hoel’s Boolean networks; cosine similarities for SBMs and CA |
| C6 | The SVD lumping commutes with the dynamics for any grouping | false | Proposition 5; counter-example below |
Method
SVD of P; choose ε; P̃ = P V_{N×r}; k-means on the rows of P̃ into r groups (Φ); stationary flow F = diag(μ)P, F′ = ΦᵀFΦ, P′ the row-normalized F′.
Concepts
- Dynamical reversibility: invertibility of P with a stochastic inverse; not the usual time reversibility of a stationary chain.
- Approximate dynamical reversibility Γ_α, and its per-state average γ_α = Γ_α/N.
- Information pathway: a row of the TPM, read as a channel.
- Clear and vague causal emergence: rank-deficiency, and near-rank-deficiency at a stated ε.
Connections
It builds on Hoel et al. 2013 (LIT-1038, THEORY-222) and Hoel 2017. It answers Eberhardt and Lee’s criticisms of EI maximization (ambiguity and non-commutativity, cited [11], not registered) with the 3×3 example of B.3. It names Rosas et al.’s information decomposition (LIT-1244) and Barnett and Seth’s dynamical independence (THEORY-257) as alternatives tied to the data distribution. Hoel’s CE 2.0 (LIT-1136, S3) adapts it by dropping σ₁, and Jansma and Hoel (LIT-1123) cite it as needing to be “expanded” to CE 2.0. Thibeault et al.’s low-rank hypothesis is cited as related.
Bearing on the record
- CLAIM-164. The LIT says the paper’s singular-value form is close to the claim’s gaps. It is, and the reading sharpens how. The paper’s coarse-graining keeps the span of the top r_ε singular vectors and only then clusters, so its object is the subspace, as CLAIM-164 says it must be. But its definitions do not require a gap: Definition 4 gives vague emergence for any ε at or above σ_N and below σ₁, so every TPM whose singular values are not all equal shows it at some ε, and the degree varies with ε. The paper picks ε at “relatively clear cut-offs”; that is CLAIM-164’s criterion, used informally and not stated. Proposed dated note for CLAIM-164 below.
- QUESTION-115. The LIT filed it as the 2024 candidate “on searching for coarse-grainings rather than assuming them”. It is the reverse: it avoids the search by reading the spectrum, and recovers a partition afterwards by k-means. It is a criterion for whether a state space is compressible, not for which variables form a system.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- Proposition 5 is false. It claims P·Φ = Φ·P′ “for arbitrary clustering projection Φ”, and the proof uses (Φᵀ)⁻¹, which does not exist: Φ is N × r. Two states lumped together must have equal block sums of their rows for any P′ to satisfy it. A record check on the B.3 matrix, made irreducible by setting its third row to (0.1, 0.1, 0.8), with grouping {1,2},{3}, gives P·Φ rows (0.9, 0.1) and (0.8, 0.2) for the two lumped states, which no Φ·P′ can reproduce. The paper’s own B.3 shows the same non-commutativity for the EI-optimal grouping. The stationary-flow lumping preserves the stationary distribution, not the dynamics.
- Proposition 1’s proof writes an invertible P as UΛU⁻¹ with U orthonormal, which assumes P is diagonalizable by a unitary. The result itself is a standard one; the argument has a gap.
- Vague emergence is ε-relative, so “independent of coarse-graining” holds for clear emergence only.
- State spaces, not variables. The authors list this, and the difficulty of estimating TPMs from data, as weak points (§3).
- Slips. Proposition 3’s α range differs from its proof’s; the proof of Theorem 1 cites Lemma 9 for Lemma 11; Lemma 6 lists the singular values as √n₁, √n_r, …, √n_r; Fig. 4’s caption assigns “α = 0, β = 1” to two different groups of micro-states; one lemma is cited as “Lemma ??”. Code is available only on request.
Open questions
- Is there a principled ε, for instance the widest gap in the spectrum, that makes vague emergence a property of P alone?
- With a lumping that does commute (exact lumpability), how do ∆Γ and the EI gain compare?
Note of 2026-10-29: Eberhardt and Lee, registered and read
Eberhardt and Lee 2022, which Connections gives as cited [11] and not registered, is now registered and read: LIT-1839, NOTE-1413. The reading narrows the criticism this paper answers. The partitions in their (13) and (17) are strongly lumpable, so in (17) the coarse-graining commutes with time evolution and only the EI-optimal choice changes with the time step; the ambiguity charge stands on (14). Their non-commutativity is a failure of the selection criterion, EI maximization, not of the lumping. Proposition 5, found false above, concerns the lumping itself, which commutes only when the rows of lumped states have equal block sums; that is the strong lumpability (17) has. This note’s second open question, how ∆Γ and the EI gain compare for a lumping that does commute, is not answered by that reading, which computes no ∆Γ.