Field Memory: Formal Specification

Abstract We present Field Memory as a mathematically precise associative memory module. The work provides a rigorous specification of a key-value memory architecture represented as a complex-valued matrix that superposes multiple associations within a single memory field. Earlier informal descriptions are revised and clarified. In particular, unsupported claims such as constant-time retrieval are corrected, yielding a retrieval complexity of O(n) per stored item. The paper derives the formal key-value binding and retrieval equations, analyzes interference effects and storage capacity using established associative memory models, and provides explicit numerical examples illustrating the behavior of the system. Field Memory represents knowledge items as complex vectors ψ ∈ Cⁿ whose components take the form ψᵢ = Aᵢ e^{iϕᵢ}. Each stored element consists of a key vector ψ_k and a value vector ψ_v. Associations are encoded using outer-product binding, producing matrices β_{kv} = ψ_k ψ_v* that are added together through linear superposition. The resulting memory matrix M = Σ ψ_k ψ_v* stores all associations simultaneously. Retrieval is performed by presenting a probe vector ψ_q and computing a correlation operation ψ̂_v = M† ψ_q, producing a value estimate weighted by the similarity between the query and stored keys. The work situates Field Memory within the broader class of hetero-associative memory systems, demonstrating its formal equivalence to classical outer-product associative memory models including Hopfield networks and distributed memory frameworks. The paper explicitly analyzes the effect of interference caused by superposition of multiple stored patterns. For random keys in dimension n, cross-talk noise grows proportionally to m/n, where m is the number of stored associations. This yields a practical capacity scaling of O(n) items for a fixed retrieval error tolerance. These bounds are consistent with known results from holographic reduced representations and Hopfield memory capacity analyses. The architecture is further examined in the context of Retrieval-Augmented Generation pipelines. A Morphological Mapper encodes documents and queries into complex semantic vectors that form key-value pairs. These pairs are stored in the Field Memory matrix, and retrieval produces a blended value vector which can be used by a generative model to produce answers without explicit text concatenation. The paper describes the encoder-memory-decoder pipeline and discusses the implications of replacing explicit retrieval indices with a superposed associative representation. Worked examples illustrate both ideal retrieval under orthogonal keys and noisy retrieval under realistic conditions with random vectors and perturbations. Complexity analysis compares the approach with modern approximate nearest neighbor search systems such as HNSW and FAISS. While Field Memory provides an interpretable linear associative operator, its storage capacity and retrieval complexity are significantly more constrained than large-scale vector indexing systems. The work emphasizes that the contribution of Field Memory lies in providing a mathematically explicit framework for associative retrieval within RAG systems rather than proposing a production-scale retrieval infrastructure. Limitations regarding capacity, deletion, update complexity, and numeric precision are discussed in detail. The paper concludes by outlining directions for empirical validation, including simulation studies of interference behavior, benchmarking against ANN retrieval systems, and experimental integration with generative models.