Source
Rosas et al. (2024), arXiv:2402.09090, LIT-1306, read in full in NOTE-1030: Definition 7, Lemma 4 (cited from Geiger and Temmel 2014), Proposition 4, Theorem 3 and Counterexamples 4, 6 and 7. Lumpability is from Kemeny and Snell (Finite Markov Chains, §6.3–6.4). The link to column-equitable partitions is the record’s, from THEORY-755 (Shpak et al., LIT-1242). The derivations below are the record’s.
What was actually shown
X is a Markov chain on a countable state space with transition matrix P (rows are origins). The blocks C₁, C₂, … partition the states, and Z_t = f(X_t) is the block label. Write P(x → C) = Σ_{y∈C} P(x, y).
Strong lumpability is a row condition. Kemeny and Snell define it as: Z is Markov, with the same transitions, for every initial law. Suppose P(x → C_I) = q(J, I) for every x ∈ C_J (the row condition). Then p(Z_{t+1} | X⃗_t) = p(Z_{t+1} | X_t) = q(Z_t, ·), so Z is Markov with q from any start. Conversely, start from a point mass at x ∈ C_J: P(Z₁ = I | Z₀ = J) = P(x → C_I) must equal q(J, I) for every such x.
It is THEORY-755’s column-equitability, for the master equation. The law evolves as p_{t+1} = Pᵀp_t. For A = Pᵀ, the column sum of block I over column j is Σ_{i∈C_I} A_ij = P(j → C_I). Column-equitability asks that it be the same for every j ∈ C_J, which is the row condition. So “the block occupation probabilities evolve autonomously from every initial law” (THEORY-755, with point masses among the initial states) is strong lumpability.
Strong lumpability ⟹ informational closure. By the row condition, p(Z_{t+1} | X⃗_t) = q(Z_t, ·), a function of Z_t. So I(X⃗_t; Z_{t+1} | Z⃗_t) = 0, and by THEORY-820 (a) Z is informationally closed, i.e. dynamically independent. No irreducibility, aperiodicity or stationarity of the start is needed. The paper’s Proposition 4 assumes all three.
Closure with a Markov label ⟹ strong lumpability, if X is irreducible.
- Take the stationary law π. If X is irreducible and positive recurrent (as every finite irreducible chain is), then π(x) > 0 for every x.
- Closure gives H(Z_{t+1} | Z⃗_t) = H(Z_{t+1} | X⃗_t) = H(Z_{t+1} | X_t). Markovity of Z gives H(Z_{t+1} | Z⃗_t) = H(Z_{t+1} | Z_t). So I(Z_{t+1}; X_t | Z_t) = 0.
- Then p(z′ | x) = p(z′ | f(x)) for every x of positive mass, and hence for every x. That is the row condition.
- Aperiodicity is not used. The paper’s footnote 109 leaves open whether it can be dropped.
The Markov hypothesis is needed. Let X_t = (Z_t, Z_{t−1}) for a binary second-order Markov Z with all transitions positive. X is an irreducible, aperiodic Markov chain. X_t is a function of Z⃗_t, so Z is closed. It is not strongly lumpable, because p(Z_{t+1} | X_t) depends on Z_{t−1}. The paper’s Counterexample 6, a deterministic 4-cycle, makes the same point with a periodic chain.
Weak lumpability does not give closure. Counterexample 4 (checked). Under the uniform stationary law Z is i.i.d., hence Markov, yet the transfer entropy is 1 bit. Weak lumpability is closure of the label’s law from one initial law. Informational closure needs the micro state to add nothing, whatever the micro state is.
Second-order lumpability does not give closure either. Counterexample 7, checked: H(Z_{t+1} | Z_t, Z_{t−1}) = 0.7329, H(Z_{t+1} | X_t) = 0.5521, and the transfer entropy is 0.1808 bits at both 2 and 6 steps of history.
A numerical check of the forward direction. A three-state chain with P(a → c) = P(b → c) = 0.3, lumped as {a, b}, {c}: transfer entropy 0 at one to four steps of history. With P(b → c) moved to 0.2, it is 0.0064 bits.
On the causal-state chain (Theorem 3). Let E_t be the causal states of any stationary process X, a Markov chain with unifilar transitions.
- (4) Strong lumpability gives closure. Suppose E is strongly lumpable under f, and set Z_t = f(E_t). The law of X_{t+1} given X⃗_t depends only on E_t, and E_{t+1} is fixed by E_t and X_{t+1}. So p(Z_{t+1} | X⃗_t) = p(Z_{t+1} | E_t) = q(Z_t, ·), and Z is informationally closed. Z is then a coarse-graining of X’s past, not in general of its present state.
- (3) Closure gives strong lumpability. Let Z = f(X_t) be spatial and informationally closed. By THEORY-820, E′t = f*(E_t), with E′{t+1} = δ(E′t, Z{t+1}) by unifilarity. So p(E′{t+1} | E_t) = Σ_z p(Z{t+1} = z | E_t)·1[δ(E′t, z) = ·]. Here p(Z{t+1} | E_t) = p(Z_{t+1} | X⃗_t) = p(Z_{t+1} | Z⃗_t) = p(Z_{t+1} | E′t), by closure. So p(E′{t+1} | E_t) depends on E_t only through E′_t. That is the row condition, and E is strongly lumpable under f*, for every causal state of positive mass.
- (1)–(2) Weak lumpability. Computational closure, E′ = f*(E), makes E weakly lumpable under f*, because E′ is Markov under the stationary law. Conversely, if f(E) is Markov under the stationary law, its causal states are a function of f(E_t), hence of E_t. The paper’s definition asks for Markov “for at least one initial distribution”. The proof needs the stationary one.
What could have failed: closure of the label could have needed more than a row condition, such as conditions on how mass is spread inside blocks. It does not. The within-block distribution matters only for weak lumpability, which is the wrong notion for closure.
What this does not say
- Not that the closed coarse-grainings form a lattice. Strongly lumpable partitions are closed under common coarsening: within a block of the join, states are linked by chains of shared blocks of either partition, and each link preserves the probabilities into the join’s blocks. They are not closed under common refinement (THEORY-820’s three-bit example, where A and B are strongly lumpable and (A, B) is not).
- Not anything approximate. A partition whose row sums nearly agree is not shown to be nearly closed. One example in NOTE-1030 (a jittered modular walk) fits that, and it is no bound.
- Not that closure needs modularity. On a random walk with uniform weights, every partition is strongly lumpable (checked on 50 random partitions of eight nodes into two or three groups). The paper’s modular walk closes for every ratio r_b/r_w, including r_b > r_w (checked at 0.01, 1 and 3).
- Not closure of a single block’s indicator. The label of the whole partition closes. Membership in one block need not.
Connections
- THEORY-755. For a Markov chain’s master equation, its condition is strong lumpability. This entry adds that the block label is then dynamically independent, which THEORY-755 showed only for the linear Gaussian VAR reading.
- THEORY-733. Its L A = B L is the Gaussian counterpart of the row condition: p(Y_{t+1} | X_t) depends on X_t only through Y_t. There, with Σ ≻ 0, closure forces it without assuming Y is Markov. Here a closed label can have memory, because a chain can carry exact delayed copies (X_t = (Z_t, Z_{t−1})), which non-degenerate noise rules out.
- CLAIM-388. Uniform coupling makes every partition aggregable in the VAR. Uniform weights make every partition strongly lumpable in the walk. In neither is any group a weakly coupled block.
- CLAIM-389. Counterexample 4 is its discrete counterpart: a label that is Markov in its own law and still dynamically dependent.
- QUESTION-218: answers in part. The paper’s closed units are partitions of the state space. In most of its examples these are the orbits of a permutation symmetry: counts, energy levels, overlaps, module labels. That is the equitable-partition kind of unit THEORY-755 added, not a block.
Note of 2026-10-29: two strongly lumpable partitions in Eberhardt and Lee
Eberhardt and Lee, LIT-1839, read in NOTE-1413, give two strongly lumpable partitions without calling them that. In their (13), every one of states 1–7 sends all its mass into the block {1–7}. In their (17), both rows of {x₁, x₂} send 0.5 into each block. For (17) the reading computed that A(T)² equals the {x₁, x₂} coarsening of T², the matrix [[0.75, 0.25], [0.5, 0.5]]: the coarse-graining commutes with time evolution, as this entry’s lumpability guarantees. What their example shows failing is effective information’s choice of partition, not the coarse-graining.
Pfante et al. 2014, LIT-1837, which Rosas et al. (NOTE-1030) credit with the equivalence this entry generalizes for Markov spatial coarse-grainings, is registered and Unavailable, so the record still holds it second-hand. The entry is unchanged.