The claim
The kernels are fixed by C. Project onto the span of the current value Y_t. Taking the covariance of the discrete Mori–Zwanzig equation (THEORY-819) with Y₀, the noise term drops out, being orthogonal to the projection space at the initial time, and what remains is the Volterra recursion C(m+1) = Σ_{k=0}^{m} c_k C(m−k). In generating functions, 1 − Σ_k c_k w^{k+1} = C(0)/Ĉ(w). So the kernels c_k are a functional of C, that is, of Y’s own second-order law. The memory (c_k for k ≥ 1) vanishes iff C(m) = c₀^m C(0).
Its order. Let Ĉ = N/D in lowest terms with D(0) = 1, so that the Hankel rank of C is d = max(deg D, deg N + 1) (Kronecker). The memory sequence has generating function num/N, with w²·num = N − C(0)D − c₀wN. num and N share no root: a common root r would give C(0)D(r) = 0, against coprimality. So the memory’s Hankel rank is max(deg N, deg num + 1). If deg N = d − 1, that is d − 1. Otherwise deg D = d, the w^d coefficient of N − C(0)D − c₀wN is −C(0)D_d ≠ 0, so deg num = d − 2, and the rank is again d − 1.
Checks (NOTE-1029). In a random n = 4 VAR(1) the Mori kernels computed from the projection blocks matched the Volterra recursion from C to 2 × 10⁻¹⁶, with d = 4 and memory rank 3; seven cases in all, including CLAIM-389’s counterexample (d = 1, rank 0). Re-run by the coordinator.
What it does not say
- Not that memory detects dependence. Since the kernels depend only on Y’s own law, no memory measure can tell whether the rest of the system drives Y. That needs the joint law, which is what THEORY-257’s transfer entropy uses (CLAIM-389).
- Not about other projections. Projecting onto Y’s whole past (the Wiener projection) gives a different kernel with the same numerator and a different denominator in the VAR(1) case; that is in NOTE-1029, not here.
- Nothing about nonlinear functionals. Second order only.
Note of 2026-10-21: the widening to any functional of the law
A5 of the exchange of 2026-10-20 restated this claim’s “What it does not say” as “Proposition A”: “No method operating exclusively on the marginal law of the observed process can, in general, identify its dynamical independence from omitted variables.” It cited this claim without noticing that it already says so for memory. A4 had put it as “whether external influence can be detected in the reduced process’s own temporal structure”, and A7 §III made it a proposition: “This is not a sample-size limitation. It survives even with perfect knowledge of the observable distribution.”
The widening, from memory measures to any functional of Y’s law, is shown by one pair, checked on 2026-10-21. CLAIM-389’s Y has autocovariance 0.5ᵗ and unit variance. A stationary Gaussian process is fixed by its mean and autocovariance, so Y has the law of the autonomous AR(1) Y′t = 0.5 Y′{t−1} + η_t with Var η = 0.75. Yet T(X → Y) = ½ ln(0.75/0.59) = 0.120 nats per step, while Y′, taken as its own whole system, receives nothing. By CLAIM-350, a property decidable from 𝓛(Y) must be constant on the models that share that law, and dynamical independence is not. Dependence can still be found from the joint law of the whole system, and nothing here is about interventions.
The pair was proposed as a claim of its own and not filed; it is recorded here and on CLAIM-389. QUESTION-252’s leave-one-out uses it as the witness for memory without independence.