Wave-Based Semantic Memory: Concepts & Mechanisms

• Wave-based semantic memory is a framework where semantic content is represented as complex waveforms with amplitude indicating salience and phase conveying context.
• It employs retrieval methods based on resonance and interference, leveraging spectral decomposition to structure and access distributed semantic information.
• Architectural implementations include phase-coded RAG systems and traveling-wave models, demonstrating practical benefits in operator-sensitive retrieval and efficient memory utilization.

Searching arXiv for directly relevant papers on wave-based semantic memory, phase-coded semantic retrieval, traveling-wave memory, and adjacent semantic-memory models. Searching for "Wave-Based Semantic Memory" on arXiv. Searching for papers on traveling-wave memory and semantic memory on arXiv. Wave-based semantic memory denotes a family of proposals in which semantic content is represented not only as a static point in a real-valued embedding space, but as a phase-bearing, spectral, or propagating pattern whose retrieval depends on resonance, interference, or wave-like dynamics. In the most direct formulations, a memory item is encoded as a complex waveform

$\psi(x)=A(x)e^{i\phi(x)},$

with amplitude $A(x)$ interpreted as semantic intensity or salience and phase $\phi(x)$ as contextual or operator-level structure; retrieval is then driven by constructive or destructive interference rather than cosine similarity alone (Listopad, 21 Aug 2025). Other work extends the same intuition to holographic field memory for retrieval-augmented generation (Saklakov, 14 Nov 2025), DFT-based semantic hierarchy construction (Kasubuchi et al., 16 Feb 2026), complex-valued token representations with shared global magnitude and token-specific phase (Zhang et al., 2024), traveling-wave substrates for working memory (Karuvally et al., 2024), and biological pattern-wave mechanisms in cortex (Redozubov, 2014). The field is therefore heterogeneous: some papers propose explicit semantic-wave formalisms, whereas others provide adjacent mechanisms that are wave-like, phase-aware, or distributed but are not themselves full theories of semantic memory.

1. Conceptual scope and mathematical primitives

The most explicit definition of wave-based semantic memory treats knowledge as a complex-valued pattern rather than a real vector. In this formulation, each stored item has the form

$\psi(x)=A(x)e^{i\phi(x)},$

where $A(x)\ge 0$ is semantic amplitude and $\phi(x)\in[-\pi,\pi)$ is contextual phase. This division is central: amplitude carries salience, while phase carries contextual modulation, operator-level semantics, and distinctions such as negation or discourse shift (Listopad, 21 Aug 2025). A closely related architectural proposal adopts the same core object and describes memory as a distributed semantic field in which meaning is “smeared across the dimensions” and stored as holographic traces,

$\psi_k(x)=A_k(x)e^{i\phi_k(x)}.$

Retrieval is then understood as phase-sensitive resonance rather than nearest-neighbor lookup over isolated vectors (Saklakov, 14 Nov 2025).

A second mathematical idiom recasts semantic structure in spectral terms. WavePhaseNet applies a DFT along the sequence dimension of an embedding matrix $V\in\mathbb{R}^{d\times T}$,

$$\tilde{V}_{n,k}=\sum_{t=0}^{T-1} V_{n,t} e^{-2\pi i kt/T},$$

and interprets low-frequency components as global meaning and intent, with high-frequency components carrying local syntax and expression. A reduced representation is reconstructed as

$$V_{\text{reduced}}=\mathrm{IDFT}\bigl(\tilde{V}[:,\Omega]\bigr),$$

where $A(x)$0 is a selected low-frequency band (Kasubuchi et al., 16 Feb 2026). This is a wave-based semantic memory in a different sense: semantics are not phase-coded at the level of single memory items, but organized as spectral bands with different abstraction levels.

A third formulation appears in the Wave Network, which represents each token as a complex vector

$A(x)$1

Here $A(x)$2 is shared across the whole sequence and is intended to represent global semantics, while $A(x)$3 is token-specific and encodes each token’s relation to that global semantic state (Zhang et al., 2024). This is not a memory system in the database sense, but it is a wave-based semantic representation in which local and global meaning are separated into magnitude and phase.

Taken together, these formalisms support a general characterization: wave-based semantic memory is any semantic-memory scheme in which meaning depends on amplitude-phase structure, frequency decomposition, or propagating field dynamics rather than on real-valued similarity alone. This suggests a common emphasis on distributed encoding, phase-sensitive discrimination, and nonlocal interaction, even when the specific mechanisms differ.

2. Storage, superposition, and semantic organization

The direct storage model in phase-aware semantic memory stores items as amplitude-phase pairs rather than materialized complex arrays. ResonanceDB, the implementation associated with the phase-aware retrieval framework, stores fixed-length patterns as real-valued pairs $A(x)$4, converts them to complex form on demand, and performs exact scan-based retrieval with complexity $A(x)$5 over $A(x)$6 patterns of length $A(x)$7. The reported implementation uses memory-mapped binary segments, a deterministic comparison kernel, a scalar CPU backend called JavaKernel, and an experimental SIMD backend called SimdKernel (Listopad, 21 Aug 2025). This is significant because it makes wave-based semantic memory a concrete storage substrate rather than a purely conceptual analogy.

The field-memory formulation is more explicitly superpositional. The Field Memory Layer stores each knowledge item as a complex waveform and superposes them into a shared field:

$A(x)$8

A stored memory is therefore a holographic trace distributed across the whole field rather than a discrete slot. Retrieval occurs by injecting a query waveform into that field and amplifying traces whose phase-frequency structure matches the query (Saklakov, 14 Nov 2025). This is a stronger claim than phase-aware similarity alone: memory is not merely a collection of complex vectors, but a standing semantic field.

WavePhaseNet organizes memory hierarchically through frequency selection and consistency constraints. The paper assumes spectral energy

$A(x)$9

defines cumulative retained energy

$\phi(x)$0

and argues that for $\phi(x)$1, preserving about $\phi(x)$2 of the energy yields $\phi(x)$3 low-frequency components (Kasubuchi et al., 16 Feb 2026). It then supplements this spectral core with overlapping local windows $\phi(x)$4, local sections $\phi(x)$5, and graph-based consistency penalties. The resulting Semantic Conceptual Hierarchy Structure is not a symbolic ontology, but a distributed semantic hierarchy in which global semantic traces are low-frequency and local semantic fragments are glued together by cohomological regularization.

A much more speculative physical theory, “Is Semantics Physical?!,” also presents semantics as a distributed dynamical structure with a dual encoding. Under the hypothesis of boundedness, semantic units are represented both as specific symbol sequences and as the performance of associated engines; hierarchical semantic organization is said to be stabilized by matter waves and non-local feedback in spatially distributed reaction-diffusion systems (Koleva, 2010). The paper is explicit that semantic units are inter-basin orbits and that reproducibility is supported by the additive decomposition of the power spectrum into a discrete band and a continuous $\phi(x)$6 band. This is unconventional and highly abstract, but it extends wave-based semantic memory beyond complex embeddings toward physical dynamics.

3. Retrieval by resonance, interference, and phase manipulation

The defining retrieval operator in phase-aware semantic memory is the resonance score

$\phi(x)$7

with

$\phi(x)$8

Using

$\phi(x)$9

the score can be written as

$\psi(x)=A(x)e^{i\phi(x)},$0

The central point is that retrieval depends on the real part of a complex correlation, hence on $\psi(x)=A(x)e^{i\phi(x)},$1, so anti-phase dimensions suppress similarity rather than disappearing into a phase-insensitive norm (Listopad, 21 Aug 2025).

This retrieval rule has several formal properties: $\psi(x)=A(x)e^{i\phi(x)},$2, symmetry, self-match $\psi(x)=A(x)e^{i\phi(x)},$3 for $\psi(x)=A(x)e^{i\phi(x)},$4, anti-phase minimum $\psi(x)=A(x)e^{i\phi(x)},$5 when $\psi(x)=A(x)e^{i\phi(x)},$6, and global phase invariance $\psi(x)=A(x)e^{i\phi(x)},$7 (Listopad, 21 Aug 2025). In the special case of purely real patterns with equal norms,

$\psi(x)=A(x)e^{i\phi(x)},$8

so resonance reduces to a shifted cosine-like similarity only in that restricted setting. This is why the framework presents itself as a phase-aware alternative rather than a rebranding of vector similarity.

Negation and operator-sensitive retrieval are modeled as phase transformations. The strongest formal case is anti-phase:

$\psi(x)=A(x)e^{i\phi(x)},$9

which yields destructive interference and a zero resonance score (Listopad, 21 Aug 2025). The field-memory architecture generalizes this by treating morphology as a source of phase operations. A negating prefix is described as adding approximately $A(x)\ge 0$0 to relevant semantic phases, while retrieval selects

$A(x)\ge 0$1

from the field (Saklakov, 14 Nov 2025). This is the basis of “morphological-semantic resonance”: semantic operators act directly on phase rather than only on amplitude or vector position.

The reported empirical result for this family is narrow but important. On a compact operator set built from the bases “happy” and “good,” resonance achieves $A(x)\ge 0$2 for NEG, SHIFT+, INT_UP, and INT_DOWN, whereas cosine yields $A(x)\ge 0$3 across these operators in that setup (Listopad, 21 Aug 2025). The paper also reports distinct resonance-distance bands for NEG, SHIFT+, INT_UP, and INT_DOWN, whereas cosine distances collapse near the same value. This does not establish broad semantic superiority, but it does show a direct advantage on operator-sensitive retrieval.

WavePhaseNet approaches retrieval differently. Instead of itemwise resonance, it treats reasoning as reconstruction and harmonization: low-frequency global intent $A(x)\ge 0$4 is coupled to local sections $A(x)\ge 0$5 with

$A(x)\ge 0$6

while local inconsistency is penalized by

$A(x)\ge 0$7

The semantic-memory analogue is that retrieval is a globally consistent reassembly of distributed semantic fragments rather than a direct resonance ranking (Kasubuchi et al., 16 Feb 2026).

4. Architectural realizations and computational substrates

The most complete end-to-end architecture is the phase-coded RAG proposal built from three tiers: a Morphological Mapper, a Field Memory Layer, and a Non-Contextual Generator. The mapper transforms text $A(x)\ge 0$8 into a complex waveform $A(x)\ge 0$9, the field memory stores superposed traces $\phi(x)\in[-\pi,\pi)$0, and the generator performs iterative retrieval-conditioned decoding by repeatedly formulating a probe, reading a resonant vector from memory, integrating it into state, and producing the next token (Saklakov, 14 Nov 2025). The proposal is explicit that it is a conceptual architecture rather than a fully validated implementation, but it is important because it turns wave-based semantic memory into a full retrieval-generation loop.

The Wave Network offers a smaller and more empirically grounded substrate for wave-style semantics. Each token is represented as

$\phi(x)\in[-\pi,\pi)$1

where $\phi(x)\in[-\pi,\pi)$2 is the global semantic magnitude obtained by an $\phi(x)\in[-\pi,\pi)$3-aggregation over the sequence, and $\phi(x)\in[-\pi,\pi)$4 encodes the token’s relation to that global semantic vector. Two update operators are defined: interference by complex addition and modulation by complex multiplication. On AG News, the single-layer Wave Network reaches $\phi(x)\in[-\pi,\pi)$5 accuracy with wave interference and $\phi(x)\in[-\pi,\pi)$6 with wave modulation; the paper states that this approaches a fine-tuned BERT base model at $\phi(x)\in[-\pi,\pi)$7 while using a 2.4-million-parameter model, and reports reductions in video memory usage and training time relative to BERT base during wave modulation (Zhang et al., 2024). This is not a semantic memory store, but it is a concrete demonstration that amplitude-phase representations can support efficient semantic computation.

Traveling-wave working memory provides an adjacent dynamic substrate. The model stores an $\phi(x)\in[-\pi,\pi)$8-step history of $\phi(x)\in[-\pi,\pi)$9-dimensional states in a rectangular neural lattice $\psi_k(x)=A_k(x)e^{i\phi_k(x)}.$0 and updates it by a shift plus boundary injection:

$\psi_k(x)=A_k(x)e^{i\phi_k(x)}.$1

$\psi_k(x)=A_k(x)e^{i\phi_k(x)}.$2

The paper proves that any history-dependent dynamical system with state dimension $\psi_k(x)=A_k(x)e^{i\phi_k(x)}.$3, history $\psi_k(x)=A_k(x)e^{i\phi_k(x)}.$4, and evolution function $\psi_k(x)=A_k(x)e^{i\phi_k(x)}.$5 can be represented in this traveling-wave model, and shows that the linear boundary-condition case yields a state-transition operator that is effectively a shift matrix plus a low-rank boundary update (Karuvally et al., 2024). The model concerns working memory rather than long-term semantic memory, but it matters because it supplies a mathematically explicit example of information being stored as a propagating field rather than fixed slots.

5. Adjacent models, misconceptions, and boundary cases

Not every distributed or dynamical account of semantic memory is wave-based. “Switcher-random-walks” formulates semantic retrieval as a Markovian network search with local clustering and nonlocal switching,

$\psi_k(x)=A_k(x)e^{i\phi_k(x)}.$6

and evaluates exploration by mean first passage time. The paper is explicit that it does not propose a wave equation, oscillatory semantic field, or literal wave-propagation account of semantic memory; it tracks a single walker on a concept network rather than a distributed activation field (0903.4132). It is therefore a complementary search model, not evidence for semantic waves.

Sparsey is similarly adjacent. It stores episodic traces as sparse distributed representations in superposition and argues that semantic memory emerges as a side effect of storing episodic traces whose overlaps preserve higher-order statistical structure (Rinkus et al., 2017). This is strongly relevant to distributed semantic memory, but its superposition is combinatorial and population-based rather than phase-based or wave-mechanical. There are no oscillatory fields, complex amplitudes, or interference equations. The model is best read as a bridge between overlap-based semantic structure and later wave-based proposals.

Streaming semantic-memory systems can also be semantically organized without being wave-based. SAVEMem builds a three-tier memory for streaming video under a constant budget, scores visual tokens using a fixed pseudo-question bank

$\psi_k(x)=A_k(x)e^{i\phi_k(x)}.$7

and performs query-aware late-interaction retrieval over short-, mid-, and long-term memory (Wu et al., 8 May 2026). Its relevance is architectural rather than mechanistic: it separates write-time semantic retention from read-time query retrieval, a design principle that could transfer to wave-based memory even though the model uses ordinary hidden states and cosine similarity rather than waves.

A further source of confusion is terminological. In physics, “gravitational wave memory” refers to the permanent effect obtained by integrating a radiative field that satisfies a wave equation with divergence-form source terms. The observable is a permanent displacement of freely falling masses, not a semantic-memory process (Garfinkle, 2022). The phrase “wave memory” in that literature is conceptually unrelated to semantic memory, despite the shared vocabulary. For wave-based semantic memory, the relevant meanings of “wave” are phase-coded semantic patterns, traveling neural activity, or spectral decomposition, not gravitational-wave memory.

6. Neurobiological interpretations, limitations, and open directions

The strongest neurobiological wave-memory proposal in this corpus is the pattern-wave model of brain function. It argues that extrasynaptic metabotropic receptor clusters on dendrites detect local spatial patterns of nearby activity generated by diffuse neurotransmitter spillover, and that any compact pattern of neural activity can emit a diverging wave of endogenous spikes whose wave-front spike pattern is strictly unique to the initiating pattern (Redozubov, 2014). In this framework, memory is not primarily a synaptic-weight matrix; it is a distributed system of receptor-cluster states and reproducible propagating waves. The paper is highly relevant to wave-based memory mechanisms, but it does not provide a standard theory of semantic memory in terms of concepts, lexical structure, or compositional semantics.

“Is Semantics Physical?!” extends the biological-dynamical direction even further. Under boundedness, semantics is said to emerge spontaneously from non-Markovian constrained transitions, inter-basin orbits, and matter-wave-mediated non-local feedback in reaction-diffusion systems. Semantic units are claimed to have an exclusive two-fold representation as symbol sequences and engine performances, and hierarchical semantic organization is stabilized by emitted matter waves (Koleva, 2010). The paper is explicitly unconventional, sparse in proofs, and programmatic rather than operational, but it remains one of the few works that directly links semantic organization to wave-mediated physical stabilization.

The limitations of the field are correspondingly clear. The phase-aware retrieval framework explicitly notes the absence of a learnable phase modulator, sensitivity to phase noise and high-frequency perturbations, exact scan-based retrieval with $\psi_k(x)=A_k(x)e^{i\phi_k(x)}.$8 complexity, amplitude-calibration issues, and experimental latencies reported on a moderate-scale exact-search prototype rather than on ANN indices (Listopad, 21 Aug 2025). The field-memory RAG proposal presents no full benchmark section, no direct comparison to strong RAG baselines, and no rigorous interference-capacity analysis (Saklakov, 14 Nov 2025). WavePhaseNet is largely theoretical, with underdeveloped empirical validation and several malformed equations in its presentation (Kasubuchi et al., 16 Feb 2026). Traveling-wave memory establishes a wave substrate for bounded history, but not a model of durable concept-level knowledge (Karuvally et al., 2024).

A plausible implication is that wave-based semantic memory is presently best understood not as a single settled model, but as a research program organized around several recurring hypotheses: semantic content may require phase as well as magnitude; retrieval may be better modeled as resonance or interference than as purely geometric proximity; semantic hierarchy may be spectrally organized; and distributed wave-like substrates may ease memory access or binding in ways that static vector stores obscure. What remains unresolved is how to learn phase systematically, how to control interference at scale, how to represent durable conceptual structure rather than finite-history traces, and how to connect wave-style representations to broad semantic tasks beyond operator-sensitive retrieval and controlled memory experiments (Listopad, 21 Aug 2025).