One Spectrum, Two Resources: Data-Memory Scaling in Autoregressive Prediction

How much learned memory is needed to benefit from more data? We show that the two resources are governed by one predictive-energy spectrum in a positive-entropy autoregressive retrieval source. Each coordinate contributes its query probability times the squared radius of its unknown logit. Writing \(\mu\) for the resulting energy spectrum, we prove the minimax law ${\mathfrak{R}}_{\mathrm{value}}^{\ast }(n,B){\asymp }_R{\Phi }_{\mu }(n^{-1})+{\Phi }_{\mu }({\tau }_B),{\Phi }_{\mu }(t)=\int \min \{x,t\}\mu (\mathrm{d}x),$ for \(n\) prediction blocks and a learned state with at most \(2^B\) values. Data set the resolution \(1/n\); memory sets the level \(\tau_B\) reached by optimal bit allocation. The complete curve also recovers the positive spectrum. Energy-dimension pairing is essential: two causal sources with identical block-energy and block-dimension marginals have different data and memory exponents. A masked query-key attention head learns the route and values, realizing the law with explicit routing, format, and arithmetic errors. Further results give exponent-adaptive allocation, finite-precision realization, and compute-precision laws under two-sided arithmetic assumptions. Experiments recover the data-memory collapse and coupling exponents, explain the routing and allocation mechanisms, and examine weight-only quantization across six pretrained-model scales.