Causal Memory Kernel

• Causal memory kernel is defined as a temporal weighting function whose support is restricted to past states, ensuring retarded dependence in evolution equations.
• It appears in diverse settings—from generalized quantum master equations to stochastic point-process models and heat conduction—each leveraging its causal structure.
• Key insights include the application of Hardy-space analyticity, Kramers–Kronig relations, and the emergence of non-Hermitian effects that challenge conventional spectral analysis.

Searching arXiv for the cited papers and related ^^^^^^^^10causal memory kernel10^^^^^^^^ work. arxiv_search.query({"^^^^^^^^10search_query10 OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^^^^^"," OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10^^^^,"^^^^^^^^10sort_by10 arxiv_search.query returned ^^^^10submittedDate10^^^^ papers. Searching for additional arXiv papers explicitly using the phrase "^^^^^^^^10causal memory kernel10^^^^^^^^". arxiv_search.query({"^^^^^^^^10search_query10 memory kernel10^^^^^^^^\""," OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10^^^^,"^^^^^^^^10sort_by10 A ^^^^^^^^10causal memory kernel10^^^^^^^^ is a temporal weighting function, scalar- or operator-valued, whose support is restricted to the past and which therefore enters evolution equations only through retarded dependence on prior states or prior noise. In stationary form this is often written as PRESERVED_PLACEHOLDER_^^^^10search_query10^^^^ for PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^, or equivalently PRESERVED_PLACEHOLDER_^^^^10max_results10^^^^; in non-Markovian open quantum dynamics it appears as the Nakajima–Zwanzig kernel PRESERVED_PLACEHOLDER_^^^^10sort_by10^^^^ governing the generalized quantum master equation. Recent arXiv work develops this notion in several distinct directions: as a vector-valued Hardy-space object satisfying Kramers–Kronig relations in open quantum systems, as a non-Hermitian kernel whose temporal correlations can be invisible to power spectra, as an explicitly causal kernel that violates the fading-memory paradigm in heat conduction, and as a functional-analytic building block for nonlinear adaptive memory (&&&^^^^10search_query10^^^^&&&, &&&^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^&&&, &&&^^^^10max_results10^^^^&&&, &&&^^^^10sort_by10^^^^&&&).

^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^. Formal definitions and common structural features

Across the cited literature, the defining condition is retarded support. On a compact interval PRESERVED_PLACEHOLDER_^^^^10submittedDate10^^^^, a stationary memory kernel is specified by

PRESERVED_PLACEHOLDER_^^^^10sort_order10^^^^

so that its two-time action is PRESERVED_PLACEHOLDER_^^^^10descending10^^^^ for PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10^^^^. In point-process modeling, the same condition appears as

PRESERVED_PLACEHOLDER_^^^^10search_query10^^^^

with PRESERVED_PLACEHOLDER_^^^^10all:\10^^^^ for PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10^^^^. In transport theory, causal memory is encoded by an integral over PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^, and in open quantum dynamics it is encoded by the retarded propagator PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10max_results10^^^^ inside the exact kernel construction (&&&^^^^10sort_by10^^^^&&&, &&&^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^&&&, &&&^^^^10max_results10^^^^&&&, &&&^^^^10search_query10^^^^&&&).

These formulations differ in codomain and interpretation. Some kernels are scalar and enter constitutive laws; some are operator-valued and act on reduced density operators; some are paired with a state-dependent sensitivity factor. What remains common is that the present state depends on a weighted history, not solely on instantaneous data.

Setting	Kernel or memory object	Causal condition
Open quantum GQME	PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10sort_by10^^^^	retarded evolution in PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10submittedDate10^^^^
Stationary stochastic process	PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10sort_order10^^^^	PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10descending10^^^^ for PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10causal memory kernel10^^^^
Nonlinear adaptive memory	PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10^^^^	PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10all:\10^^^^ for PRESERVED_PLACEHOLDER_^^^^10max_results10search_query10^^^^
Heat conduction	PRESERVED_PLACEHOLDER_^^^^10max_results10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^	integral restricted to PRESERVED_PLACEHOLDER_^^^^10max_results10max_results10^^^^

A useful conceptual distinction follows immediately. Causality specifies temporal support; it does not, by itself, determine whether memory decays, grows, oscillates, or changes sign. That distinction becomes central in the later developments.

^^^^10max_results10^^^^. Nakajima–Zwanzig kernels in non-Markovian open quantum dynamics

In the open-quantum-system setting, the ^^^^^^^^10causal memory kernel10^^^^^^^^ is the central object of the exact generalized quantum master equation for the reduced state PRESERVED_PLACEHOLDER_^^^^10max_results10sort_by10^^^^:

PRESERVED_PLACEHOLDER_^^^^10max_results10submittedDate10^^^^

Here

PRESERVED_PLACEHOLDER_^^^^10max_results10sort_order10^^^^

with total Hilbert space PRESERVED_PLACEHOLDER_^^^^10max_results10descending10^^^^, Liouvillian PRESERVED_PLACEHOLDER_^^^^10max_results10causal memory kernel10^^^^, and projection PRESERVED_PLACEHOLDER_^^^^10max_results10search_query10^^^^, PRESERVED_PLACEHOLDER_^^^^10max_results10all:\10^^^^. The Laplace transform,

PRESERVED_PLACEHOLDER_^^^^10sort_by10search_query10^^^^

is the frequency-domain object on which analyticity and Kramers–Kronig analysis are performed (&&&^^^^10search_query10^^^^&&&).

The same framework distinguishes three related objects: the memory kernel, the reduced-state propagator, and the effective kernel. Their distinction matters because they have different sensitivity to initial system-bath correlations. A factorized initial state,

PRESERVED_PLACEHOLDER_^^^^10sort_by10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^

enforces PRESERVED_PLACEHOLDER_^^^^10sort_by10max_results10^^^^ and hence PRESERVED_PLACEHOLDER_^^^^10sort_by10sort_by10^^^^, yielding the causal homogeneous GQME. By contrast, when initial correlations are present, the reduced state remains analytic in the upper half-plane for any initial state, whereas the effective kernel obeys only a perturbative modified Kramers–Kronig relation with upper-half-plane corrections at zeros of the reduced-state propagator. In the solvable Jaynes–Cummings example with finite Fock truncation, force-fitting correlated dynamics into a homogeneous GQME produces

PRESERVED_PLACEHOLDER_^^^^10sort_by10submittedDate10^^^^

together with upper-half-plane poles and a doubled KK-residual PRESERVED_PLACEHOLDER_^^^^10sort_by10sort_order10^^^^, from PRESERVED_PLACEHOLDER_^^^^10sort_by10descending10^^^^ to PRESERVED_PLACEHOLDER_^^^^10sort_by10causal memory kernel10^^^^, indicating apparent acausality induced by discarded initial correlations (&&&^^^^10search_query10^^^^&&&).

^^^^10sort_by10^^^^. Hardy-space analyticity and Kramers–Kronig structure

K. Liu’s analysis places the Laplace-transformed memory kernel in a vector-valued Hardy framework under explicit bath hypotheses. The assumptions are a thermodynamic limit in which the spectrum of PRESERVED_PLACEHOLDER_^^^^10sort_by10search_query10^^^^ is purely continuous on PRESERVED_PLACEHOLDER_^^^^10sort_by10all:\10^^^^, with coupling-weighted spectral density PRESERVED_PLACEHOLDER_^^^^10submittedDate10search_query10^^^^, and a local PRESERVED_PLACEHOLDER_^^^^10submittedDate10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^ regularity condition PRESERVED_PLACEHOLDER_^^^^10submittedDate10max_results10^^^^ for some PRESERVED_PLACEHOLDER_^^^^10submittedDate10sort_by10^^^^. Under these conditions and factorized initialization,

PRESERVED_PLACEHOLDER_^^^^10submittedDate10submittedDate10^^^^

defines an analytic operator-valued function on the upper half-plane, and for every PRESERVED_PLACEHOLDER_^^^^10submittedDate10sort_order10^^^^,

PRESERVED_PLACEHOLDER_^^^^10submittedDate10descending10^^^^

Equivalently, PRESERVED_PLACEHOLDER_^^^^10submittedDate10causal memory kernel10^^^^ (&&&^^^^10search_query10^^^^&&&).

This Hardy placement yields rigorous Kramers–Kronig relations for the boundary values PRESERVED_PLACEHOLDER_^^^^10submittedDate10search_query10^^^^:

PRESERVED_PLACEHOLDER_^^^^10submittedDate10all:\10^^^^

If PRESERVED_PLACEHOLDER_^^^^10sort_order10search_query10^^^^, the once-subtracted form is used:

PRESERVED_PLACEHOLDER_^^^^10sort_order10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^

The supporting causal statement is the operator-valued analogue of the Titchmarsh or Paley–Wiener theorem: support of PRESERVED_PLACEHOLDER_^^^^10sort_order10max_results10^^^^ in PRESERVED_PLACEHOLDER_^^^^10sort_order10sort_by10^^^^ implies upper-half-plane analyticity and KK relations, and conversely KK analyticity implies PRESERVED_PLACEHOLDER_^^^^10sort_order10submittedDate10^^^^ for PRESERVED_PLACEHOLDER_^^^^10sort_order10sort_order10^^^^.

Three physical consequences are highlighted. First, the CPTP–Hardy consistency criterion states that if a finite-dimensional rational approximation PRESERVED_PLACEHOLDER_^^^^10sort_order10descending10^^^^ develops an isolated pole with PRESERVED_PLACEHOLDER_^^^^10sort_order10causal memory kernel10^^^^, then the corresponding propagator PRESERVED_PLACEHOLDER_^^^^10sort_order10search_query10^^^^ acquires at least one pole in the upper half-plane, implying exponential growth of PRESERVED_PLACEHOLDER_^^^^10sort_order10all:\10^^^^ and therefore non-physical dynamics because any CPTP map satisfies PRESERVED_PLACEHOLDER_^^^^10descending10search_query10^^^^. Second, for passive bosonic baths satisfying

PRESERVED_PLACEHOLDER_^^^^10descending10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^

the kernel belongs, after sign reversal, to the operator-valued Herglotz–Nevanlinna class, so dissipative kernels automatically satisfy KK. Third, a Carleman criterion for MKCT moments,

PRESERVED_PLACEHOLDER_^^^^10descending10max_results10^^^^

guarantees that diagonal Padé PRESERVED_PLACEHOLDER_^^^^10descending10sort_by10^^^^ approximants converge locally uniformly in the upper half-plane and preserve the KK relations. In the independent-boson model with Drude–Lorentz bath, the coherence-channel prediction PRESERVED_PLACEHOLDER_^^^^10descending10submittedDate10^^^^ holds to machine precision under small initial correlations; in the Jaynes–Cummings model, approximately PRESERVED_PLACEHOLDER_^^^^10descending10sort_order10^^^^ of coherence-channel zeros of PRESERVED_PLACEHOLDER_^^^^10descending10descending10^^^^ lie in the open upper half-plane, with PRESERVED_PLACEHOLDER_^^^^10descending10causal memory kernel10^^^^ (&&&^^^^10search_query10^^^^&&&).

^^^^10submittedDate10^^^^. Non-Hermitian causal memory and the limits of spectral analysis

A distinct construction appears in a stochastic point-process model with strictly causal memory. The kernel is written

PRESERVED_PLACEHOLDER_^^^^10descending10search_query10^^^^

with shape function

PRESERVED_PLACEHOLDER_^^^^10descending10all:\10^^^^

and PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10search_query10^^^^. The instantaneous rate is then

PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^

with white-noise driving field PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10max_results10^^^^ satisfying PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10sort_by10^^^^, followed by Poisson sampling PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10submittedDate10^^^^ (&&&^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^&&&).

Because PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10sort_order10^^^^, the associated effective operator

PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10descending10^^^^

is non-Hermitian. The resulting rate correlation for PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10causal memory kernel10^^^^ is

PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10search_query10^^^^

and the power spectral density satisfies

PRESERVED_PLACEHOLDER_^^^^10causal memory kernel10all:\10^^^^

The central claim is not merely that the process is causal, but that its causal-memory signature can be invisible to conventional spectral methods. In particular, the Gaussian contribution centered at PRESERVED_PLACEHOLDER_^^^^10search_query10search_query10^^^^ transforms into a broad factor PRESERVED_PLACEHOLDER_^^^^10search_query10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^, with no sharp spectral line at PRESERVED_PLACEHOLDER_^^^^10search_query10max_results10^^^^. The paper therefore identifies a limitation of spectral analysis for non-Hermitian, non-stationary processes.

The same model predicts a sharp transition in similarity space at PRESERVED_PLACEHOLDER_^^^^10search_query10sort_by10^^^^, with asymmetry

PRESERVED_PLACEHOLDER_^^^^10search_query10submittedDate10^^^^

and orientation dependence

PRESERVED_PLACEHOLDER_^^^^10search_query10sort_order10^^^^

with PRESERVED_PLACEHOLDER_^^^^10search_query10descending10^^^^. The reported quantitative agreement is PRESERVED_PLACEHOLDER_^^^^10search_query10causal memory kernel10^^^^ with PRESERVED_PLACEHOLDER_^^^^10search_query10search_query10^^^^, while the power spectra exhibit no peak at PRESERVED_PLACEHOLDER_^^^^10search_query10all:\10^^^^. This combination of causal construction, non-Hermitian asymmetry, and Fourier silence is central to the article’s notion of observable temporal correlations that evade standard spectral diagnostics (&&&^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^&&&).

^^^^10sort_order10^^^^. Causal kernels that violate fading memory

In heat-conduction theory, Herrera proposed a causal kernel that explicitly contravenes the fading-memory paradigm. The constitutive law is

PRESERVED_PLACEHOLDER_^^^^10all:\10search_query10^^^^

with

PRESERVED_PLACEHOLDER_^^^^10all:\10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^

Here PRESERVED_PLACEHOLDER_^^^^10all:\10max_results10^^^^ is the thermal relaxation time and PRESERVED_PLACEHOLDER_^^^^10all:\10sort_by10^^^^ is a constant with units of conductivity divided by time; for simplicity, PRESERVED_PLACEHOLDER_^^^^10all:\10submittedDate10^^^^ is chosen so that the short-time limit recovers a coefficient of order PRESERVED_PLACEHOLDER_^^^^10all:\10sort_order10^^^^ (&&&^^^^10max_results10^^^^&&&).

The kernel is causal because it is supported only for PRESERVED_PLACEHOLDER_^^^^10all:\10descending10^^^^, but it is not fading in the usual sense. Relative to the Cattaneo–Vernotte kernel PRESERVED_PLACEHOLDER_^^^^10all:\10causal memory kernel10^^^^, it assigns negligible weight to the instantaneous gradient and larger weight to gradients in the remote past. This makes the example especially important conceptually: causal support does not require monotone decay of memory weight.

Coupling the constitutive law to energy conservation produces a third-order-in-time evolution equation for the temperature. In the compact notation of the paper,

PRESERVED_PLACEHOLDER_^^^^10all:\10search_query10^^^^

where PRESERVED_PLACEHOLDER_^^^^10all:\10all:\10^^^^ and PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10search_query10^^^^. The model supports damped oscillatory modes and, in the applications discussed there, yields a quasi-periodic exchange between heating and cooling before relaxation rather than the monotonic behavior associated with Fourier’s law or the single damped oscillator of Cattaneo’s law. The paper applies this transport equation to thermohaline convection and nuclear burning instability, and it also sketches a relativistic generalization by modifying the gradient-memory term in Israel–Stewart theory (&&&^^^^10max_results10^^^^&&&).

^^^^10descending10^^^^. Nonlinear adaptive memory and state-dependent sensitivity

A further generalization replaces purely time-weighted memory by memory modulated by the evolving state itself. On a fixed compact horizon PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^, the functional-analytic framework for nonlinear adaptive memory classifies kernels into three hierarchical classes: mathematically admissible kernels PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10max_results10^^^^, regular admissible kernels PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10sort_by10^^^^, and generalized admissible kernels PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10submittedDate10^^^^. For PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10sort_order10^^^^ the defining properties are positivity, normalization,

PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10descending10^^^^

uniform bounds PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10causal memory kernel10^^^^, and a Lipschitz condition. For PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10search_query10^^^^ the requirements are PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10all:\10^^^^ and PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10^^^^ control together with bounded variation and a nonvanishing total integral magnitude (&&&^^^^10sort_by10^^^^&&&).

State dependence enters through an adaptive sensitivity function PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^ satisfying bounds

PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10max_results10^^^^

Lipschitz continuity in the state argument,

PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10sort_by10^^^^

measurability in PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10submittedDate10^^^^, and positivity at the zero trajectory. A concrete history-dependent construction is

PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10sort_order10^^^^

which interpolates between instantaneous response at PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10descending10^^^^ and genuine historical feedback. The key Lipschitz estimate is

PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10causal memory kernel10^^^^

The associated adaptive memory-dependent functional is

PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10search_query10^^^^

with adaptive memory space

PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10all:\10^^^^

The framework establishes absolute convergence, measurability, uniform boundedness, positive definiteness, and comparison with the supremum norm:

PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10max_results10search_query10^^^^

If the maximum of PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10max_results10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^ is attained at an interior point, then

PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10max_results10max_results10^^^^

so the memory contribution is strictly nontrivial. The theory also shows that PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10max_results10sort_by10^^^^ is strictly contained in PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10max_results10submittedDate10^^^^; in particular, discontinuous functions such as the indicator PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10max_results10sort_order10^^^^ belong to the memory space, capturing abrupt on-off switching and other jump-like signals (&&&^^^^10sort_by10^^^^&&&).

^^^^10causal memory kernel10^^^^. Conceptual distinctions and diagnostics

Several recurring misconceptions are clarified by the combined literature. First, causality is not synonymous with fading memory. Herrera’s kernel is fully causal yet gives greater weight to remote-past gradients than to recent ones (&&&^^^^10max_results10^^^^&&&). Second, the absence of a sharp power-spectrum feature does not imply the absence of causal memory. The non-Hermitian point-process model produces statistically significant temporal structure in similarity space while remaining spectrally silent at the characteristic scale (&&&^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^&&&). Third, apparent acausality in an extracted kernel need not reflect underlying noncausal dynamics; in the open-quantum example it can arise from force-fitting dynamics with initial correlations into a homogeneous GQME, producing upper-half-plane poles and enlarged KK residuals (&&&^^^^10search_query10^^^^&&&).

Taken together, these results suggest that “^^^^^^^^10causal memory kernel10^^^^^^^^” is best understood as a structural condition on temporal support, to which additional properties—Hardy analyticity, KK consistency, passivity, non-Hermiticity, anti-fading weighting, or state-dependent sensitivity—are appended according to the physical or mathematical setting. Within open quantum dynamics, the strongest current formulation is the correspondence

PRESERVED_PLACEHOLDER_^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10max_results10descending10^^^^

while in stochastic, transport, and nonlinear functional settings the same causal-support principle organizes markedly different phenomena and diagnostics (&&&^^^^10search_query10^^^^&&&, &&&^^^^10id:(Liu, 18 Apr 2026) OR id:(Pinheiro, 30 Mar 2026) OR id:(L et al., 2019) OR id:(Jiang, 4 Apr 2026)10^^^^&&&, &&&^^^^10max_results10^^^^&&&, &&&^^^^10sort_by10^^^^&&&).