THEORY-819: For an invertible map with an invariant measure, projecting onto the closed span of the whole past of chosen functions of the resolved variables (a Wiener projection) turns the exact discrete Mori–Zwanzig equation into one with no memory term: the next resolved value is a causal linear filter of those past functions plus a residual orthogonal to all of them, so the memory is carried by the coarse variable's own history; a finite NARMAX form follows only by a rational approximation of the filter, uncontrolled in general but exact for a scalar coarse variable of a stable linear Gaussian VAR(1)

Source

Lin and Lu (2021), Journal of Computational Physics 424:109864, LIT-1307, read in full in NOTE-1029 from arXiv v5. Steps 1–5 are the paper’s Eqs. 2.2–2.6 and 3.3–3.10, re-derived here. Step 6 is the record’s.

What was actually shown

Let X_{n+1} = F(X_n) with F invertible and μ an F-invariant probability measure, X₀ ~ μ. Let H = L²(μ) and Mφ = φ∘F (the Koopman operator), x = π(X) ∈ ℝᵈ the resolved variables, and Ψ(x) = [ψ₁(x) … ψ_ν(x)] chosen observables.

Step 1: the Dyson formula. For any linear operator P with Q = I − P, M^{n+1} = Σ_{k=0}^{n} M^{n−k}PM(QM)^k + (QM)^{n+1}. For n = 0 this is M = PM + QM. If it holds for n, multiply on the left by M. The sum’s terms become M^{n+1−k}PM(QM)^k, and the remainder becomes M(QM)^{n+1} = PM(QM)^{n+1} + (QM)^{n+2}. Its first part is the k = n + 1 term of the sum for n + 1, and its second part is the new remainder.

Step 2: the discrete MZ equation. Let P be a projection (PQ = 0) whose range consists of functions of x. Apply Step 1 to π and evaluate at X₀. (M^{n+1}π)(X₀) = x_{n+1}, and (M^{n−k}g)(X₀) = g(X_{n−k}). Put ξ_k = (QM)^kπ, so M(QM)^kπ = ξ_k∘F, and P(ξ_k∘F) is a function Γ_k of x. Then x_{n+1} = PF(x_n) + Σ_{k=1}^{n} Γ_k(x_{n−k}) + ξ_{n+1}(X₀), with Pξ_k = PQ(Mξ_{k−1}) = 0 for k ≥ 1. With P the orthogonal projection onto span(Ψ), ξ_n is orthogonal to Ψ(x₀) only.

Step 3: M is unitary. ⟨Mu, Mv⟩ = ∫(u∘F)(v∘F)dμ = ∫uv dμ by invariance, so M is an isometry. It is onto because F is invertible (u = (u∘F⁻¹)∘F), so M⁻¹ = M*.

Step 4: the collapse. Let W be the closed span of {M^{−k}ψ_i : k ≥ 0, i ≤ ν}. M^{−k}ψ_i = ψ_i∘F^{−k}, so W is spanned by the observables at every past time. M⁻¹ maps the spanning set into itself and is bounded, so M⁻¹W ⊆ W, i.e. Q_W M⁻¹P_W = 0. Taking adjoints, with (M⁻¹)* = M: P_W M Q_W = 0. Then for k ≥ 1, P_W M(Q_W M)^k = (P_W M Q_W)M(Q_W M)^{k−1} = 0, and Step 1 reduces to M^{n+1} = MⁿP_W M + (Q_W M)^{n+1}. Applied to π at X₀: x_{n+1} = (P_W(π∘F))(X_n) + ξ_{n+1}(X₀). The first term is an element of W composed with Fⁿ: a limit of finite combinations Σ_k Ψ(x_{n−k})·h_k. When it is a convergent series, this is the paper’s Eq. 3.6a. There is no memory term, because the past is inside the projection.

Step 5: the residual is orthogonal to the whole past. P_W M Q_W = 0 means M maps W⊥ into W⊥, so MQ_W = Q_W M Q_W. Hence (Q_W M)² = (Q_W M Q_W)M = MQ_W M, and by induction (Q_W M)^{n+1} = MⁿQ_W M. So ξ_{n+1} = ξ₁∘Fⁿ with ξ₁ = Q_W(π∘F) ∈ W⊥. For m ≤ n, by unitarity, ⟨ξ_{n+1}, ψ_i∘F^m⟩ = ⟨ξ₁, ψ_i∘F^{m−n}⟩ = 0, since ψ_i∘F^{−(n−m)} ∈ W. The residual at n + 1 is orthogonal to Ψ(x_m) for every m ≤ n, past the initial time included. ⟨ξ_m, ξ_n⟩ and ⟨ξ_m, Ψ(x_n)⟩ depend only on m − n. That orthogonality is the normal equation of the causal Wiener filter: the first term is the minimum mean-square linear prediction of x_{n+1} from the entire past of Ψ.

Checked numerically (NOTE-1029). The Dyson formula held to 3 × 10⁻¹⁶ on random 12 × 12 matrices. For an orthogonal M with a 7-dimensional M⁻¹-invariant W, P_W M Q_W was 1 × 10⁻¹⁵ and the collapsed formula held to 3 × 10⁻¹⁵. For a non-normal M with an M⁻¹-invariant W, P_W M Q_W = 1.58 and the collapsed formula failed by 2.8, so unitarity is doing real work.

Step 6: the rational form is exact in the linear Gaussian VAR(1) class (the record’s). Take X_{t+1} = AX_t + ε_t, A stable, ε Gaussian white with Σ ≻ 0, as a skew product (Step 3 then applies), with scalar Y = LX and Ψ(y) = y.

1. By CLAIM-398, φ(B)Y = u with φ(w) = det(I − Aw) and u an MA of order at most n − 1.
2. On the unit circle Y’s spectral density is L(I − Ae^{−iω})⁻¹Σ(I − Ae^{−iω})^{−*}Lᵀ ≥ λ_min(Σ)‖L(I − Ae^{−iω})⁻¹‖², and this is positive. So u’s spectrum |φ|²S_Y is a positive trigonometric polynomial. By Fejér–Riesz, u = θ(B)e, with θ(0) = 1, deg θ ≤ n − 1, every root of θ strictly outside the unit disk, and e white.
3. Y = (θ/φ)(B)e is causal, and e = (φ/θ)(B)Y has geometrically decaying coefficients. So the closed span of Y’s past equals that of e’s past, and e_{t+1} is orthogonal to it.
4. Hence Y_{t+1} = [1 − φ(B)/θ(B)]Y_{t+1} + e_{t+1}, and the first term lies in the span of the past. It is therefore the Wiener projection, and e is the residual of Step 5.
5. Since θ(0) = φ(0) = 1, 1 − φ/θ = w·b(w)/θ(w) with deg b ≤ n − 1. The filter’s generating function Σ_k h_k w^k is b(w)/θ(w): exactly the paper’s B/A, with both degrees at most n − 1.
6. Its poles are the reciprocals of θ’s roots, inside the unit circle. So the paper’s decaying-memory condition holds, and h_k decays at the largest uncancelled MA zero.

Checked: h from a 300-lag Toeplitz solve agreed with the series of b/θ to 1 × 10⁻¹⁶ – 3 × 10⁻¹³ in four models (random n = 3 and n = 4, QUESTION-222’s example sampled at h = 0.1, CLAIM-389’s counterexample), and the residual variance equalled the innovation variance (1.139424, 13.605055, 0.097536, 0.750000). In QUESTION-222’s example h₀ = 0.97628 and h_k falls by 0.90088 per lag, the MA zero.

What the paper could have shown otherwise. With a non-unitary M, or with W not invariant under M⁻¹ (a window of the past rather than all of it), P_W M Q_W ≠ 0 and the memory terms return. Step 4’s numerical counterexample shows the first.

What this does not say

• Not a finite augmentation with controlled error, in general. The exact reduced equation uses the entire past. The paper’s NARMAX is a rational approximation it calls “uncontrolled” and “heuristic”. It bounds no error, proves no convergence in the orders, and chose its KS and Burgers orders by trial. Only in the linear Gaussian class of Step 6 is the finite form exact.
• Not that memory is removed. It is moved into the projection. The Mori kernels (projection onto the present) and the Wiener filter (projection onto the past) of the same process differ, as A3 cautioned. In the linear class they share the numerator det(I − Aw) and differ in the denominator, det(I − K₂₂w) against θ(w) (NOTE-1029).
• Not independence. The projections are onto linear spans, so the residual is uncorrelated with every past Ψ, not independent of the past. With conditional expectation in place of the linear projection, the paper’s footnote states, and it is immediate, that the residuals are martingale differences. Nothing here says that is enough for a white-noise NARMAX.
• Not that a convergent series always exists. Step 4 gives an element of a closed span. The paper’s property (i) asserts coefficients h_k, unique under linear independence, which needs more than it states. Step 6 supplies the series in the linear class.
• Not that this is a mark of a level. Every stationary coarse variable has such an own-history representation in second order. What THEORY-257 tests is whether the micro past predicts the residual.
• Nothing for non-invertible maps or non-stationary data. The paper keeps both assumptions, and the collapse fails without unitarity.

Connections

• QUESTION-222: answers the “exact” half, an exact reduced equation built from the coarse variable’s own history. It leaves open the “finite, with controlled error” half outside linear classes.
• CLAIM-398: the ARMA state of that claim is a finite realization of this Wiener filter. Step 6 identifies the filter’s poles with the MA zeros. That is why NOTE-1028 found the lag-truncation error falling at θ² per lag.
• CLAIM-389: its kernels M_j are the paper’s Γ_k for the finite-rank projection. NOTE-1029 shows that they, like the Wiener filter, are fixed by Y’s autocovariance alone.
• THEORY-257: the Wiener projection conditions on exactly the coarse variable’s own past. In the linear Gaussian case the Wiener residual variance is the numerator of the transfer entropy (0.75 in CLAIM-389’s counterexample).
• LIT-1304 and THEORY-818: Chorin and Lu’s NARMAX, which this paper reinterprets as an approximate Wiener projection, with the Mori–Zwanzig link now stated exactly rather than by analogy.
• NOTE-989 and NOTE-975: the continuous-time MZ readings. There the noise is orthogonal to the present resolved variables only, which is the finite-rank case of Step 2.