GM-SAGE: Graph-based Multi-bounce SAGE Algorithm

• The paper introduces a graph-constrained adaptation of the SAGE algorithm that replaces direct parameter search with a dictionary-based search over scatterer coordinates and bounce sequences.
• It separates one-bounce, two-bounce, and higher-bounce components via iterative updates and graph-based pruning to improve estimation accuracy in near-field and blocked scenarios.
• Simulation results demonstrate reduced localization errors and robust performance under spatial non-stationarity and partial blockage in complex indoor and ELAA environments.

Searching arXiv for the specified GM-SAGE and related SAGE papers to ground the article with current records. Graph Theory-based Dictionary-aided Multi-bounce SAGE (GM-SAGE) is a graph-constrained adaptation of the space-alternating generalized expectation-maximization (SAGE) algorithm for estimating and localizing multi-bounce propagation in near-field sensing and environment reconstruction. In the reported formulation, graph theory is used to model multi-bounce propagation of the near field, indoor reflectors and scatterers are represented as vertices in a propagation graph, and multi-bounce paths are represented by edges linking these vertices (Liu et al., 16 Sep 2025). GM-SAGE is “dictionary-aided” because it builds candidate channel atoms from graph-consistent scatterer coordinates and path sequences rather than from decoupled range and angle parameters, and it is “multi-bounce” because one-bounce, two-bounce, and higher-bounce components are explicitly separated and alternately updated within a SAGE procedure (Liu et al., 16 Sep 2025). A later extension applies the same framework to environment reconstruction under array partial blockage and spatial non-stationary (SNS) channels in extremely-large antenna arrays (ELAA), where the algorithm estimates a sparse spatial amplitude field together with graph-consistent path geometry (Liu et al., 16 Sep 2025). Although the acronym “GM-SAGE” is also naturally suggested as a graph-theoretic, dictionary-aided, multi-bounce interpretation of the graph-memory retrieval architecture in SAGE for language agents, that usage is an external interpretation rather than the name introduced in the graph-memory paper (Wang et al., 12 May 2026).

1. Conceptual definition and research context

GM-SAGE was introduced to address indoor sensing under conditions where single-bounce approximations are insufficient. The motivating setting includes multi-bounce propagation, spherical wavefronts, and spatial nonstationarity caused by near-field effects, all of which complicate direct estimation of range, angle of departure (AoD), angle of arrival (AoA), and scatterer location (Liu et al., 16 Sep 2025). The central methodological move is to replace direct search over coupled geometric parameters with a graph-based search over scatterer coordinates and feasible bounce sequences.

In the propagation-graph formulation, the graph is written as $G=(V,E)$, where vertices encode scatterers or reflectors and edges encode physically viable radio links constrained by visibility, geometry, and reflection physics (Liu et al., 16 Sep 2025). Paths of bounded bounce order are then represented as walks in this graph. This permits the channel estimation problem to be reframed as a dictionary-fitting problem over graph-consistent atoms.

The term “dictionary-aided” has a precise meaning in this literature. The dictionary is parameterized by scatterer coordinates, reflectivities, velocities, and bounce order, and each dictionary atom synthesizes the full multi-element, multi-subband, multi-frame response of a candidate path under a near-field spherical-wave model (Liu et al., 16 Sep 2025). This differs from conventional SAGE treatments that search directly over range and angular parameters. In the ELAA blockage extension, the graph-based dictionary additionally restricts coordinate updates to feasible multi-bounce paths while a sparse amplitude field absorbs SNS and blockage effects (Liu et al., 16 Sep 2025).

A separate but related use of the phrase “graph theory-based, dictionary-aided, multi-bounce” emerges in the memory-retrieval framework SAGE for language agents. That system employs a dynamic heterogeneous graph, query planning, soft addressing, and structurally gated multi-hop propagation, and thus supports a GM-SAGE-style reading at the architectural level (Wang et al., 12 May 2026). However, the paper does not introduce GM-SAGE as its formal algorithmic name, nor does it include an explicit external dictionary module (Wang et al., 12 May 2026).

2. Observation model and near-field channel representation

The measurement model in “Scatterer Localization Using Multi-Bounce Paths” is specified for an $M$-Tx and $N$-Rx sensing system with $P$ non-overlapping sub-bands per frame and $Q$ frames per coherent processing interval. The baseband-equivalent measurement channel is $Z \in \mathbb{C}^{MN \times PQ}$, indexed by transmitter, receiver, sub-band, and frame (Liu et al., 16 Sep 2025). The channel is expressed as a sum of $L$ multipath components:

$$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$

where $\Delta\alpha_{m,n,l}$ models SNS across array elements, $\tau_{m,n,l}$ is the element-pair delay, and the element-wise AoD/AoA orientation vectors depend on near-field geometry (Liu et al., 16 Sep 2025).

The spherical-wave formulation is essential because AoD and AoA become range dependent for large apertures. The element-dependent distances are written as

$M$0

with total delay

$M$1

The paper explicitly notes that this spherical-wavefront model unifies near and far field; the far-field plane-wave approximation is recovered when the reference ranges are much larger than array apertures (Liu et al., 16 Sep 2025).

The ELAA partial-blockage extension adopts a related but vectorized model. There, the stacked measurement vector is $M$2, and the $M$3-th path contribution is

$M$4

where $M$5 is a binary sparsity mask indicating physical blockage per Tx-Rx pair and $M$6 models spatially varying attenuation (Liu et al., 16 Sep 2025). This explicitly separates geometric delay from the spatial amplitude field, allowing blockage-induced nulls to be estimated rather than averaged away.

A plausible implication is that GM-SAGE’s relevance increases with aperture size and environmental complexity, because both multi-bounce geometry and spatially varying amplitudes become more pronounced in those regimes. That implication is consistent with the emphasis on ELAA, spherical wavefronts, and SNS in the source papers (Liu et al., 16 Sep 2025, Liu et al., 16 Sep 2025).

3. Propagation graph and dictionary construction

The propagation graph is the structural core of GM-SAGE. In the indoor scatterer-localization formulation, vertices carry scatterer parameters including position $M$7, reflectivity $M$8, and possibly velocity $M$9, while the adjacency matrix $N$0 encodes physically feasible bounce transitions (Liu et al., 16 Sep 2025). Paths are restricted by visibility, total length, non-penetration, reflectivity, and a maximum bounce number $N$1.

A path of order $N$2 is a sequence

$N$3

with total geometric length

$N$4

and delay

$N$5

Under near-field operation, per-hop directions and element-wise orientations are determined from the geometry of these vertices rather than from independent angle parameters (Liu et al., 16 Sep 2025).

The dictionary is built from these graph-constrained paths. For bounce order $N$6, the dictionary $N$7 contains atoms synthesized from vertex sequences that satisfy the graph constraints (Liu et al., 16 Sep 2025). The estimation objective is written as

$N$8

This encodes the coupled near-field relation among range, AoD, and AoA through scatterer coordinates rather than parameterizing them separately (Liu et al., 16 Sep 2025).

The ELAA blockage formulation presents the graph abstraction in a complementary way. There, candidate scatterer hypotheses are nodes, edges encode feasible reflection connections subject to visibility and specular constraints, and a walk $N$9 corresponds to a candidate $P$0-bounce path consistent with room geometry (Liu et al., 16 Sep 2025). The associated atom is

$P$1

and the dictionary $P$2 aggregates all feasible atoms (Liu et al., 16 Sep 2025). Paths violating visibility, grazing-angle, or out-of-room constraints are pruned.

This graph-constrained dictionary performs two roles simultaneously. First, it injects geometric prior structure into the search. Second, it reduces the combinatorial burden of multi-bounce estimation by excluding infeasible path compositions before SAGE updates are performed (Liu et al., 16 Sep 2025, Liu et al., 16 Sep 2025).

4. SAGE adaptation and estimation procedure

GM-SAGE adapts SAGE by partitioning the received signal into bounce-order-specific hidden-data groups and alternating updates across those groups. In the indoor near-field paper, one-bounce, two-bounce, and higher-bounce components are denoted $P$3, $P$4, and $P$5, with associated parameter sets $P$6, $P$7, and $P$8 (Liu et al., 16 Sep 2025). The iterative formulation solves

$P$9

with residual updates

$Q$0

$Q$1

Assuming complex Gaussian noise, the incomplete-data likelihood is quadratic in the residual norm, and the effective E-step corresponds in practice to subtracting current estimates of the other bounce groups (Liu et al., 16 Sep 2025).

The key algorithmic departure from standard SAGE is the M-step parameterization. Rather than optimizing over decoupled range and AoD/AoA, GM-SAGE searches over graph vertex coordinates and feasible path sequences. Candidate atoms are evaluated, for example, through a correlation score

$Q$2

and the complex path gain for a selected atom is updated by

$Q$3

Weak atoms are pruned by thresholding $Q$4, and graph constraints suppress physically implausible paths (Liu et al., 16 Sep 2025).

In the ELAA blockage extension, the SAGE derivation is presented explicitly in standard E-step/M-step form. The hidden path-wise data satisfy

$Q$5

and the $Q$6-function decomposes as

$Q$7

Matched filtering across sub-bands gives the sufficient statistic

$Q$8

which is then used to update the equivalent amplitude field $Q$9 (Liu et al., 16 Sep 2025).

The geometry update is performed by graph-constrained residual minimization over candidate reflection coordinates:

$Z \in \mathbb{C}^{MN \times PQ}$0

The search is performed over one-bounce and two-bounce hypotheses, and bounce order is selected by minimum residual (Liu et al., 16 Sep 2025). The paper states that GM-SAGE inherits SAGE’s monotone likelihood ascent under Gaussian noise, while good initialization and graph pruning improve convergence and reduce local-minimum issues (Liu et al., 16 Sep 2025).

5. Spatial non-stationarity, partial blockage, and robustness

A major extension of GM-SAGE is its explicit treatment of spatial non-stationarity and partial blockage in ELAA systems. In that setting, the per-path spatial amplitude field is

$Z \in \mathbb{C}^{MN \times PQ}$1

where $Z \in \mathbb{C}^{MN \times PQ}$2 indicates a blocked Tx-Rx sub-channel for the $Z \in \mathbb{C}^{MN \times PQ}$3-th path (Liu et al., 16 Sep 2025). Stacking across sub-bands yields

$Z \in \mathbb{C}^{MN \times PQ}$4

with $Z \in \mathbb{C}^{MN \times PQ}$5 (Liu et al., 16 Sep 2025).

The M-step amplitude update decouples across Tx-Rx pairs and admits a matched-filter estimate

$Z \in \mathbb{C}^{MN \times PQ}$6

Empirical sparsity detection then declares blockage when $Z \in \mathbb{C}^{MN \times PQ}$7, setting $Z \in \mathbb{C}^{MN \times PQ}$8; an optional convex alternative uses an $Z \in \mathbb{C}^{MN \times PQ}$9-regularized estimate of $L$0 (Liu et al., 16 Sep 2025). The paper’s robustness rationale is that blocked entries are prevented from biasing geometry updates because the geometry search relies on the unblocked support selected through this amplitude-field estimation.

The reported blockage model is

$L$1

which makes explicit that blocked entries contribute only noise (Liu et al., 16 Sep 2025). This is a significant distinction from standard stationary-amplitude models. The paper states that standard SAGE degrades under ELAA blockage because it assumes stationary amplitudes across elements, whereas sparse-aware GM-SAGE with graph constraints maintains performance and mitigates model mismatch under SNS (Liu et al., 16 Sep 2025).

The same paper also identifies failure modes. Localization error increases when the unblocked set becomes too small or geometrically clustered; two-bounce paths remain robust but exhibit larger error under blockage than one-bounce paths; unresolved multipaths can induce apparent SNS nulls even in unblocked cases; and ambiguities such as left-to-right versus right-to-left two-bounce directionality may require beamforming, richer dictionaries, or multi-snapshot processing (Liu et al., 16 Sep 2025). These limitations are framed as consequences of model mismatch, sparse support, or insufficient geometric diversity rather than defects in the SAGE procedure itself.

6. Validation, performance, and computational characteristics

The two GM-SAGE papers report different forms of empirical validation. “Scatterer Localization Using Multi-Bounce Paths” uses measurement-calibrated ray tracing in a complex indoor office with 7×7 Tx and Rx arrays at $L$2 GHz, with multi-bounce channels generated via an open-access, measurement-calibrated ray-tracing platform and up to three bounces (Liu et al., 16 Sep 2025). The reported outcome is qualitative rather than tabulated numerically: GM-SAGE estimates one-bounce scatterer coordinates aligned with true propagation tracks and avoids the ghost scatterers produced by one-bounce-only baselines (Liu et al., 16 Sep 2025). The paper explicitly states that numeric error tables are not provided there.

“Environment Reconstruction in Multi-Bounce Channels with Array Partial Blockage” provides a more detailed simulation setup and selected localization errors. The room size is $L$3 m$L$4, with three walls and one obstacle; the ELAA has 16 Tx and 121 Rx elements with spacing $L$5; the carrier is $L$6 GHz; bandwidth is 1 GHz; sub-bandwidth is 10 MHz; $L$7 sub-bands; and SNR is 20 dB (Liu et al., 16 Sep 2025). Grid sizes are 0.1 m for one-bounce and 0.2 m for two-bounce hypotheses, and multi-bounce paths are synthesized via the Image method (Liu et al., 16 Sep 2025).

Selected localization errors reported in the blocked versus unblocked cases are concise enough to summarize directly:

Path/scenario	Blocked	Unblocked
Upper wall (one-bounce)	$L$8 m	$L$9 m
Upper$$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$0right (two-bounce), first point	$$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$1 m	$$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$2 m
Upper$$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$3right (two-bounce), second point	$$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$4 m	$$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$5 m

The same paper states that other walls and path pairs show $$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$6 m or similar performance across conditions, and that in some two-bounce configurations blocked localization is $$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$7 m higher error than unblocked (Liu et al., 16 Sep 2025). It also reports that the objective function converges to similar minima in blocked and unblocked cases, which is presented as evidence of robustness (Liu et al., 16 Sep 2025).

Computationally, the ELAA paper gives explicit per-iteration costs. Let $$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$8 be the number of Tx-Rx pairs and let $$\begin{aligned} Z_{m+(n-1)M,\, p+(q-1)P} &= \sum_{l=1}^{L} \alpha_{l}\,\Delta\alpha_{m,n,l}\,F_{\text{Rx}}\!\big(f_c, \mathbf{\Omega}_{\text{Rx},n,l}\big)\,F_{\text{Tx}}\!\big(f_c, \mathbf{\Omega}_{\text{Tx},m,l}\big) \ &\quad\times e^{-j2\pi f_p \tau_{m,n,l}}\,e^{j2\pi f_{D,l} q T_b}\,e^{j\phi_l} \;+\; w(p,q), \end{aligned}$$9 denote the number of candidates searched per path after graph pruning. Then the E-step costs $\Delta\alpha_{m,n,l}$0, the amplitude M-step costs $\Delta\alpha_{m,n,l}$1, and the geometry search costs $\Delta\alpha_{m,n,l}$2, for overall per-iteration complexity

$\Delta\alpha_{m,n,l}$3

(Liu et al., 16 Sep 2025). The near-field scatterer-localization paper likewise emphasizes search-space reduction relative to naive continuous range/AoD/AoA estimation and notes that naive path enumeration up to bounce order $\Delta\alpha_{m,n,l}$4 is $\Delta\alpha_{m,n,l}$5, but adjacency and visibility constraints substantially reduce the effective search (Liu et al., 16 Sep 2025).

A plausible implication is that GM-SAGE is most attractive when prior environmental structure is available or can be approximated well enough to build a meaningful propagation graph. Both papers rely on geometry-aware pruning and graph-constrained dictionaries to keep the estimation problem tractable (Liu et al., 16 Sep 2025, Liu et al., 16 Sep 2025).

7. Relation to graph-memory SAGE and broader interpretation

The graph-memory paper “SAGE: A Self-evolving Agentic Graph-Memory Engine for Structure-Aware Associative Memory” introduces a different system from the radio-sensing GM-SAGE literature, but it is relevant because it exhibits a formally graph-theoretic, multi-hop retrieval process that can be read as supporting a GM-SAGE-style interpretation (Wang et al., 12 May 2026). In that framework, the writer constructs a heterogeneous graph

$\Delta\alpha_{m,n,l}$6

where $\Delta\alpha_{m,n,l}$7 are entity nodes, $\Delta\alpha_{m,n,l}$8 are document or memory-fragment nodes, $\Delta\alpha_{m,n,l}$9 are entity-entity relation edges, and $\tau_{m,n,l}$0 are entity-document anchoring edges (Wang et al., 12 May 2026). The reader returns entity and document distributions together with an activated subgraph or path set.

The reader performs structured query planning, soft addressing, and structurally gated propagation. The soft-addressing score includes exact entity matches, aliases $\tau_{m,n,l}$1, type signals $\tau_{m,n,l}$2, hard constraints $\tau_{m,n,l}$3, and NER plus entity-linking signals:

$\tau_{m,n,l}$4

(Wang et al., 12 May 2026). Structurally gated propagation then modulates message passing using node features such as degree, clustering coefficient, core number, and average neighbor degree, edge-pair features such as common neighbors and Jaccard overlap, and a graph-level summary (Wang et al., 12 May 2026).

This paper explicitly states that it does not introduce or rely on an external dictionary resource as a distinct module. However, it identifies “dictionary-like mechanisms” in aliases $\tau_{m,n,l}$5, answer alias sets $\tau_{m,n,l}$6, type constraints, hard constraints, entity-linking signals, and entity descriptions used in soft addressing (Wang et al., 12 May 2026). For that reason, the graph-memory system can be described as naturally compatible with a “dictionary-aided” augmentation, but any such augmentation would be external to the paper itself (Wang et al., 12 May 2026).

The connection to GM-SAGE is therefore analogical rather than nomenclatural. In radio sensing, GM-SAGE denotes a graph-theory-based dictionary-aided multi-bounce SAGE algorithm for path and scatterer estimation (Liu et al., 16 Sep 2025, Liu et al., 16 Sep 2025). In graph memory, SAGE provides a heterogeneous graph, multi-hop associative retrieval from partial cues, structural role features, and a self-evolving writer-reader loop, all of which make a graph-theoretic, multi-bounce interpretation natural, but the paper does not define that system as GM-SAGE (Wang et al., 12 May 2026).

This distinction matters because the two lines of work share structural motifs—graph representations, path-based reasoning, constrained propagation, and iterative refinement—yet operate in different domains and optimize different observables. The radio GM-SAGE literature estimates physical propagation geometry and blockage-aware amplitude fields (Liu et al., 16 Sep 2025, Liu et al., 16 Sep 2025). The graph-memory SAGE literature optimizes evidence recovery, answer grounding, and retrieval efficiency in long-term language-agent memory (Wang et al., 12 May 2026). The shared terminology should therefore be understood as a structural correspondence rather than as evidence of a single unified formalism.