The claim
Take X_t = AX_{t−1} + ε_t, stable, ε_t ~ N(0, Σ), Σ ≻ 0, with stationary covariance Π, and a linear coarse-graining Y = LX with row space W. Work with linear functionals of the state in L²(Π), where the transition operator acts as Aᵀ and Mori’s projection is the Π-orthogonal projection onto W. Split Aᵀ into blocks: K₁₁ (W → W), K₂₁ (W → W⊥), K₁₂ (W⊥ → W) and K₂₂ (W⊥ → W⊥). The discrete-time Mori–Zwanzig decomposition then has memory kernels M_j = K₁₂K₂₂^(j−1)K₂₁, j ≥ 1 (NOTE-989).
- Independence ⇒ no memory. By THEORY-733, Y is exactly dynamically independent iff W is invariant under Aᵀ, that is iff K₂₁ = 0. Then every M_j = 0.
- No memory ⇏ independence. A = [[0.5, 0.4], [0, 0.3]], Σ = I − AAᵀ (positive definite), Y = X₁. Then Π = I, so the blocks are read directly off Aᵀ: K₂₁ = 0.4 and K₁₂ = 0, and every M_j = 0. Y’s autocovariance is 0.5ᵗ, so in its own law Y is an AR(1), Markov. Yet X₂ enters Y_t with coefficient 0.4 and is uncorrelated with Y’s whole past (Cov(X_{t−1}, X_{t−1−k}) = A^k has a zero (2, 1) entry). So T(X → Y) = ½ ln(Var(Y_t|Y⁻)/Σ₁₁) = ½ ln(0.75/0.59) = 0.120 nats. That was re-checked from the exact stationary covariances (0.1200).
In words: memory is information that leaves the coarse variable and comes back to it. Dependence is information that arrives from outside, whether or not it started inside. A coarse variable can receive news it never sent.
What it does not say
- Nothing for VAR(p ≥ 2) or for state-space models. The reading reports that there an exactly independent variable can carry memory (THEORY-733’s criterion constrains every lag, and the lags mix inside W). That was not re-checked here.
- That zero memory is common. It is not generic; the reading found it in none of 200 random models.
- Anything about nonlinear projections. For a deterministic flow with an invariant measure, the reading derives that zero memory, zero noise and exact closure coincide. That is a different statement, with no noise injected from outside, and it is recorded in the NOTE, not here.
Note of 2026-10-20: why “not conversely”
For any scalar stationary process, Mori’s memory for the projection onto the current value is a functional of the process’s own autocovariance, and vanishes iff that autocovariance is geometric (CLAIM-397, from the reading of Lin and Lu, NOTE-1029). Memory is a property of Y’s own law; dependence is a property of its joint law with the rest of the system. So no memory measure can detect dependence, which is the structural reason the converse fails.
Note of 2026-10-21: a paper’s centre, one misreading, and the AR(2) check
The exchange of 2026-10-20 (A2–A5) made this counterexample the centre of a proposed paper, twice in A2 (as “Markov without independence” and as “zero memory with transfer entropy”, one construction). A3 used it as a witness that every condition but independence holds. It is not: Y’s own past predicts Y with residual variance 0.75 against 0.59 from the full past, so predictive sufficiency holds only for prediction from Y itself.
The same law, different dependence. A5 and A7 used it as a non-identifiability result. Y’s autocovariance is 0.5ᵏ, so Y has the law of an autonomous AR(1) with φ = 0.5 and innovation variance 0.75, whose transfer entropy is 0. So no functional of Y’s law, memory measure or otherwise, decides dynamical independence. A7: “This is not a sample-size limitation.” Checked on 2026-10-21; the general form is the note of the same date on CLAIM-397.
Independent with memory. The report under “What it does not say”, that in VAR(p ≥ 2) an exactly independent variable can carry memory, was checked on 2026-10-21 in the simplest case. An autonomous AR(2), Y_t = 0.5Y_{t−1} + 0.3Y_{t−2} + ε, has Mori memory c₁ = 0.147 and c₂ = −0.0315, non-zero, with autocovariance Hankel rank 2 and memory rank 1, as CLAIM-397 predicts. So A4’s “impossible” cell (independent with memory) is impossible only in VAR(1). QUESTION-252 carries both witnesses.