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[Submitted on 24 Sep 2026] Title:The Cost of Long Memory: State, Context, and Stability Complexity in Sequence Models View a PDF of the paper titled The Cost of Long Memory: State, Context, and Stability Complexity in Sequence Models, by Yuheng Song View PDF HTML (experimental) Abstract:Long-range temporal dependence poses a resource question for sequence models: for a specified predictive-memory law, how much state, context, or dynamical criticality is required in order to forecast accurately? We study this question directly in forecasting risk. For algebraically decaying predictive memory, we prove matching upper and lower approximation bounds for exponential and finite-state modes. The best $r$-mode forecast error decays as $e^{-\Theta(\sqrt r)}$, so reaching forecast error $\tau$ needs $r=\Theta(\log^2(1/\tau))$ states or modes. Earlier curse-of-memory results establish broad limitations of stable recurrent models under different approximation notions; here both sides match for one canonical predictive target in forecast risk, which fixes the optimal resource exponent for that target. We then show that genuine fractional long memory changes the geometry itself. In particular, forecast error is measured after fractional integration, prediction from a finite context of length $L$ has an exact $1/L$ leading order, and a fixed fractional strength $d$ keeps the square-log state-complexity law. Near the short-memory boundary, we identify the relevant $d^2$ and $d^4$ scales and give a uniform constructive law in the intermediate regime. For nonlinear contextual recurrences with uniformly contractive state dynamics, we derive an exponential first-chaos envelope and an explicit necessary condition that relates forecast accuracy to the contraction margin. Vanishing forecasting error on an algebraic target forces the recurrence quantitatively toward criticality, a condition that is necessary and not by itself sufficient. Finite-sample Kullback--Leibler calculations further connect the predictive geometry to statistical information. Theorem-matched experiments with contractive state-space, gated recurrent, and attention models reproduce the state and stability predictions. Comments: 49 pages, 2 figures Subjects: Machine Learning (cs.LG) Cite as: arXiv:2610.08816 [cs.LG] (or arXiv:2610.08816v1 [cs.LG] for this version) https://doi.org/10.48550/arXiv.2610.08816 arXiv-issued DOI via DataCite Submission history From: Yuheng Song [view email] [v1] Thu, 24 Sep 2026 02:54:56 UTC (205 KB) Full-text links: Access Paper: View a PDF of the paper titled The Cost of Long Memory: State, Context, and Stability Complexity in Sequence Models, by Yuheng Song View PDF HTML (experimental) TeX Source view license Additional Features Audio Summary Current browse context: cs.LG new | recent | 2026-10 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar Loading... Data provided by: Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv (What is alphaXiv?) Links to Code Toggle CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub Toggle DagsHub (What is DagsHub?) GotitPub Toggle Gotit.pub (What is GotitPub?) Huggingface Toggle Hugging Face (What is Huggingface?) ScienceCast Toggle ScienceCast (What is ScienceCast?) Demos Demos Replicate Toggle Replicate (What is Replicate?) Spaces Toggle Hugging Face Spaces (What is Spaces?) Spaces Toggle TXYZ.AI (What is TXYZ.AI?) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower (What are Influence Flowers?) Core recommender toggle CORE Recommender (What is CORE?) IArxiv recommender toggle IArxiv Recommender (What is IArxiv?) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs. Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)