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

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arXiv:2609.13500v1 Announce Type: cross Abstract: 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 $\mathfrak R^*_{\rm value}(n,B)\asymp_R \Phi_\mu(n^{-1})+\Phi_\mu(\tau_B), \Phi_\mu(t)=\int\min\{x,t\}\,\mu(\mathrm dx),$ for $n$ prediction blocks and a learned state with at most $2^B