Boolean threshold functions, neuron capacity, and memory retrieval

Large language models demonstrate remarkable ability in factual recall, yet the fundamental limits of storing and retrieving input--output associations with neural networks remain unclear. We study these limits in a minimal setting: a linear associative memory that maps $p$ input embeddings in $\mathbb{R}^d$ to their corresponding~$d$-dimensional targets via a single layer, requiring each mapped input to be well separated from all other targets. Unlike in supervised classification, this strict separation induces~$p$ constraints per association and produces strong correlations between constraints that make a direct characterisation of the storage capacity difficult. Here, we provide a precise characterisation of this capacity in the following way. We first introduce a decoupled model in which each input has its own independent set of competing outputs, and provide numerical and analytical evidence that this decoupled model is equivalent to the original model in terms of storage capacity, spectra of the learnt weights, and storage mechanism. Using tools from statistical physics, we show that the decoupled model can store up to $p_c \log p_c / d^2 = 1 / 2$ associations, and generalise the computation of $p_c$ to linear two-layer architectures. Our analysis also gives mechanistic insight into how the optimal solution improves over a naïve Hebbian learning rule: rather than boosting input-output alignments with broad fluctuations, the optimal solution raises the correct scores just above the extreme-value threshold set by the competing outputs. These findings give a sharp statistical-physics characterisation of factual storage in linear networks and provide a baseline for understanding the memory capacity of more realistic neural architectures.

How many key-value associations can a $d\times d$ linear memory store? We show that the answer depends not only on the $d^2$ degrees of freedom in the memory matrix, but also on the retrieval criterion. In an isotropic Gaussian model for the stored pairs, we show that top-1 retrieval, where every signal must beat its largest distractor, requires the logarithmic model-size scale $d^2\asymp n\log n$. We prove that the correlation matrix memory construction, which stores associations by superposing key-target outer products, achieves this scale through a sharp phase transition, and that the same scaling is necessary for any linear memory. Thus the logarithm is the intrinsic extreme-value price of winner-take-all decoding. We next consider listwise retrieval, where the correct target need not be the unique top-scoring item but should remain among the strongest candidates. To formalize this regime, we propose the Tail-Average Margin (TAM), a convex upper-tail criterion that certifies inclusion of the correct target in a controlled candidate list. Under this listwise retrieval criterion, the capacity follows the quadratic scale $d^2\asymp n$. At load $n/d^2\toα$, we develop an exact asymptotic theory for the TAM empirical-risk minimizer through a two-parameter scalar variational principle. The theory has a rich phenomenology: in the ridgeless limit it yields a closed-form critical load separating satisfiable and unsatisfiable phases, and it predicts the limiting laws of true scores, competitor scores, margins, and percentile profiles. Finally, a small-tail extrapolation further leads to the conjectural sharp top-1 threshold $d^2\sim 2n\log n$.

The specification number $σ_n(f)$ of a Boolean threshold function $f$ on $n$ variables is the least number of points whose $f$-values determine $f$ uniquely among all threshold functions. Its essential points form the unique minimum such set. We develop Zuev's geometric interpretation: the threshold functions are the chambers of a central hyperplane arrangement in the $(n+1)$-dimensional space of weights and thresholds, and the essential points of a function correspond exactly to the facets of its chamber, so the specification number is the chamber's facet number. The lower bound $σ_n(f)\ge n+1$ becomes the fact that a pointed full-dimensional cone has at least $n+1$ facets, with equality for simplicial chambers. The average specification number $\overlineσ_n$ becomes an average facet count. We evaluate this average exactly via the resonance arrangement and bound it through a theorem of Fukuda, Tamura, and Tokuyama, obtaining $\overlineσ_n\le 2n$; hence $\overlineσ_n=Θ(n)$. This settles a question of Gutekunst, Mészáros, and Petersen. The method also extends to polynomial threshold functions. The same geometry links threshold functions with a threshold zonotope, whose vertices are modified Chow vectors. Its one-skeleton is the one-inclusion graph, and a vertex's degree is the specification number of that function. Finally, we treat the operations of Lozin et al. on functions of minimum specification number. Adding a variable and extending on a variable both take the product of a chamber closure with a half-line, preserving simpliciality. For the symmetric-variables extension we give an exact thresholdness criterion and show that minimum specification number is preserved whenever the extension is a threshold function. We also resolve a question they pose concerning a fourth operation.