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QUESTION-222: Beyond the scalar linear Gaussian case and the deterministic case of a fast, contracting eliminated variable, when a coarse variable's reduced dynamics carry memory of the eliminated variables, does it admit a finite-dimensional Markovian augmentation whose prediction error is bounded in terms of the eliminated dynamics, and is a level whose history, not its present state, does the predicting still a level?

7 min read

Properties6
StatusOpen
Version4
Tagscomplex-systems, mathematics, natural-sciences
Linedistributed-agency
Refined byQUESTION-248 — In a stable linear Gaussian VAR(1), does a scalar linear coarse variable whose reduced dynamics carry memory admit an exact finite-dimensional Markovian augmentation?, QUESTION-287 — In a deterministic singularly perturbed system whose eliminated variable is fast and contracting and whose reduced system is contracting, does the memoryless quasi-steady-state reduction predict the coarse variable with an error bounded uniformly in time in terms of the eliminated dynamics?
BearingTHEORY-818 — In two-scale Lorenz 96 without time-scale separation (ε = 0.5, K = 18, J = 20, F = 10), a discrete-time closure for each resolved variable that fits the one-step discrepancy of a fourth-order Runge–Kutta step as a NARMAX series, with one autoregressive lag, at most two lags of the resolved variable and at most one moving-average term, reproduces the mean, spread, distribution, autocorrelation and neighbour cross-correlation of x₁ (Kolmogorov–Smirnov distance 0.005) and, with observations every 0.05, extends the 20-member ensemble forecast lead time at anomaly correlation 0.6 from about 2 to about 4 time units over a polynomial-plus-AR(1) closure; the model structure was chosen by trial and error, differs between the two observation spacings, and carries no error bound, CLAIM-397 — For a scalar stationary process with autocovariance C, the Mori–Zwanzig memory kernels for the projection onto the current value are fixed by C alone, through C(m+1) = Σ_{k≤m} c_k C(m−k); the memory vanishes iff C is geometric, and when C has finite Hankel rank d the memory has Hankel rank exactly d − 1, 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), THEORY-762 — For a reversible diffusion dX = −∇U dt + √2 dW in equilibrium, an eigenfunction ψ of its generator with eigenvalue −λ is closed in conditional mean with no memory: E[ψ(X_{t+s}) | the whole past] = e^{−λs}ψ(X_t), and so the same given ψ's own past; but ψ(X_t) is a Markov process in its own right only when |∇ψ|² is a function of ψ, since otherwise its own path reveals a noise intensity that its present value does not fix, THEORY-760 — The finite-memory Mori–Zwanzig closure that Parish and Duraisamy take from Stinis, truncated at order m with one trapezoidal sub-interval and constant memory lengths, is a finite-dimensional Markovian augmentation: m + 1 auxiliary variables per resolved mode, each relaxing at rate 2/τ; at first order it replaces the memory integral by the kernel 2e^{−2s/τ₀} acting on the first-order term, which tends to τ₀ times that term, the t-model with t replaced by τ₀, when τ₀ is short against the resolved dynamics; its kernel is twice the true one at zero lag for every τ₀, so it is not exact even where the true memory is a single exponential, and nothing in the construction bounds its error, THEORY-752 — In optimal prediction of a Hamiltonian system with separable energy and canonical initial law, projected by conditional expectation, the t-model's leading-order memory term tPLQLx makes the effective Hamiltonian of the resolved variables non-increasing, so the eliminated variables' memory enters as a damping that grows linearly in time, CLAIM-389 — In a stable linear Gaussian VAR(1), exact dynamical independence of a linear coarse-graining implies that its Mori–Zwanzig memory vanishes, but not conversely: memory needs coupling out of the coarse variable and back, and dependence needs only the way back, so a coarse variable can be Markov in its own law with zero memory and still be dynamically dependent

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Why it is a question

Eliminating variables does not remove their influence. It returns as memory and noise. A5’s example: with ẋ = −ax + by and ẏ = cx − dy, keeping only x gives

ẋ(t) = −a x(t) + bc ∫₀ᵗ e^{−d(t−s)} x(s) ds + b e^{−dt} y(0),

with memory kernel K(t) = bc e^{−dt}. When d is large the memory fades and a Markovian model of x may serve. When d is small it persists. A5: “it would be premature to interpret persistent memory as evidence that the system has no meaningful macroscopic organization. It might instead mean that the correct macrostate includes history or an additional internal memory coordinate.”

A3 gave the general form, Ż = F(Z) + ∫ K(t − s, Z_s) ds + η, and a caution the record keeps: the split into drift, memory and noise depends on the projection, so “an arbitrary norm of the memory kernel is not, by itself, a representation-invariant measure of emergence.”

What the record holds

  • THEORY-257: dynamical independence conditions on Y’s own history. A5: “non-Markovianity of Z_t is not automatically a violation of its dynamical-independence criterion.” What it forbids is information from the micro history beyond that.
  • CLAIM-327: a reputation or standing is an endogenous variable because members read the accumulated past and act on it.
  • CLAIM-157 and CASE-043: a block test on synchronous dynamics misses a coupling that runs through storage, such as Otto’s notebook, and needs a time-lagged version that is not built.
  • Mori–Zwanzig, read on 2026-10-20:
    • Chorin, Hald and Kupferman (2000), NOTE-989. The decomposition is exact for any flow, and with Zwanzig’s projection the noise is uncorrelated with, not independent of, the resolved variables. For a deterministic flow with an invariant measure, zero memory, zero noise and exact closure coincide (the record’s derivation, in the NOTE).
    • Chorin, Hald and Kupferman (2002), NOTE-975. Three projections give three different splits, which bears out A3’s caution. In Hamiltonian systems the leading memory effect is a damping that grows linearly in time (THEORY-752). There is no augmentation and no error bound.
    • Parish and Duraisamy (2017), NOTE-973. The turbulence literature’s working answer has this question’s shape: the resolved modes plus (m + 1)N memory coordinates, the first of them the subgrid force (THEORY-760). Its error is not controlled: the memory length is fitted, and the kernel’s shape is fixed by a quadrature rule rather than by the eliminated spectrum.
  • Slow eigenfunctions of a reversible diffusion (THEORY-762, from Nadler et al., NOTE-986). A generator eigenfunction has E[ψ(X_{t+s}) | past] = e^{−λs}ψ(X_t), so its Mori–Zwanzig memory and noise vanish in mean, exactly. It is Markov in law only if |∇ψ|² is a function of ψ. Otherwise the memory sits in the noise intensity, and one extra coordinate, |∇ψ|², carries it. That is a finite augmentation for this class.
  • Memory against dynamical independence, settled for linear Gaussian VAR(1) (CLAIM-389). Independence implies zero memory, but zero memory does not imply independence. A5’s own example shows the gap: with c = 0 and b ≠ 0 there is no memory, yet x is not closed.
  • Germano (LIT-1251, Unavailable) supplies a related distinction: a hierarchy of filtered descriptions can exist with no closed equation at any level.
  • When the eliminated variable is fast and contracting, no augmentation is needed: the memoryless quasi-steady-state reduction predicts to O(ε) uniformly in time, with explicit constants, provided the reduced system is itself contracting (Del Vecchio and Slotine, THEORY-773; the paper’s Theorem 3 omits that hypothesis and is false without it). On A5’s example with c = κd the memory kernel tends to a delta and the Markovian error is O(1/d), checked at d = 5 to 500. Deterministic only; open for slow or noisy eliminated variables.
  • Chorin and Lu (2015, LIT-1304, NOTE-1028): a discrete NARMAX closure is a finite augmentation by past states, discrepancies and innovations. On Lorenz 96 at ε = 0.5, with no gap of time scales, one or two lags sufficed (THEORY-818, Proposed), with orders tuned by trial and no error bound. Its derivation from Mori–Zwanzig is motivation only.
  • In the linear Gaussian VAR(1) the augmentation is exact and needs only the coarse variable’s own history: a scalar coarse variable is ARMA of orders at most (n, n − 1) (CLAIM-398, demonstrated), though its Mori–Zwanzig kernels are non-zero at every lag. In one example the extra error of truncating to AR(p) fell at a rate set by the moving-average zero, not by the eliminated dynamics’ spectrum; three projections gave three memory lengths, as A3 cautioned.
  • Lin and Lu (2019, LIT-1307, NOTE-1029) prove the discrete Mori–Zwanzig identity exactly for any projection, and, for the Wiener projection onto the whole past of the resolved variables under a measure-preserving invertible map, that the memory term vanishes and the next state is a causal Wiener filter of the past (THEORY-819, Active). Their step to NARMAX is a heuristic rational approximation with no error bound. In the linear Gaussian VAR(1) the filter is exactly rational, and its pole, the moving-average zero, sets the truncation rate. Mori’s memory for a scalar variable is fixed by its own autocovariance and has Hankel rank one less than it (CLAIM-397). So every stationary variable has an own-history representation in second order; having one marks nothing. What can mark a level is that the micro past does not predict the residual (THEORY-257’s test), or that the representation is finite.

What would count as an answer

For a stated class, conditions under which the reduced process has a finite-dimensional Markovian augmentation (the coarse variable plus k memory coordinates) whose prediction error is bounded, with k and the error in terms of the spectrum of the eliminated dynamics; and a ruling on whether the record’s level criteria should be applied to the augmented state or to the instantaneous one. A5’s own limit: this is “a candidate theoretical extension, not a result already established for social or cognitive systems.”

Split, 2026-10-21 (ADR-049). The scalar linear Gaussian VAR(1) case is QUESTION-248, Answered by CLAIM-398: an exact finite augmentation from Y’s own history. This question keeps nonlinear and noisy classes, vector coarse variables, error in terms of the eliminated spectrum, and the ruling on levels.

Note of 2026-10-26: history against state, in finite chains

Three results from the Perspective Population Lab:

  • One lag suffices (v1.5, dmarx/PerspectiveLab@7e2edc2 experiments/closure_profile/). In an exact finite-chain fixture with a hidden lag, the current macrostate has a one-step defect of 0.2781 bits, and with one lag 0. A finite augmentation from the coarse variable’s own history, outside the linear Gaussian class.
  • History does not stand in for state (CASE-164). In a 192-state organization, authority’s finite-history defect falls from 0.147 bits at h = 1 to 0.132 at h = 6 and levels off, while the agents are closed with no history.
  • One target, one step (v1.9, dmarx/PerspectiveLab@9f82bfd). In a 32-member population, four lags of the six-variable macrostate cut the held-out error of the next change of the mean record by about 99.6%, and the current mean local endorsement predicts it exactly, because it is in the update law. That is not closure of the augmented state.

None is a bound in terms of the eliminated dynamics, and none is an infinite-history result. The question is unchanged.

Note of 2026-10-27: the fast contracting case split off

The status review of this date split this question again under ADR-049. When the eliminated variable is fast and contracting, and the reduced system is contracting, the memoryless reduction predicts to O(ε) for all time with explicit constants. That is QUESTION-287, Answered by THEORY-773. This question keeps what is still open: slow or noisy eliminated variables, nonlinear and vector classes, error in terms of the eliminated spectrum where memory is needed, and the ruling on levels.

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