{"paper":{"title":"Metric Non-Collapse in Learned World Models for Control: Approximation Theory, Finite-Sample Geometric Guarantees, and Deterministic Planning Transfer","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Alain Bensoussan, Minh-Binh Tran, Minh-Nhat Phung","submitted_at":"2026-08-07T14:25:33Z","abstract_excerpt":"We develop a three-part mathematical theory for metrically faithful learned world models for nonlinear deterministic control systems. First, on the approximation-theoretic side, we construct smooth exact latent realizations and verify the required finite-capacity $C^{1,1}$ approximation property for norm-constrained tensor-product B-spline classes with capacity-independent regularity budgets. Second, on the finite-sample geometric side, we introduce an encoder-only local--global metric hinge whose directional and separated-pair terms prevent infinitesimal collapse and global folding. Although "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.07265","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.07265/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}