# Finite-Sample Metric Non-Collapse for Geometrically Supervised Latent World Models in Control

arXiv: https://arxiv.org/abs/2608.07265 · HTML: https://mlsanity.com/paper/2608.07265/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Latent predictive world models, one of 11 directions in World models.
- Builds on 12 of its 36 references that are in the map.
- First posted on arXiv 7 Aug 2026.

## Placed in

- World models / [Latent predictive world models](https://mlsanity.com/wm/latent-predictive-world-models/)

## Builds on (most cited first, 12 of 12)

- [World Models](https://mlsanity.com/paper/1803.10122/) (2018)
- [V-JEPA 2: Self-Supervised Video Models Enable Understanding, Prediction and Planning](https://mlsanity.com/paper/2506.09985/) (2025)
- [When Does LeJEPA Learn a World Model?](https://mlsanity.com/paper/2605.26379/) (2026)
- [Value-guided action planning with JEPA world models](https://mlsanity.com/paper/2601.00844/) (2025)
- [Predictive but Not Plannable: RC-aux for Latent World Models](https://mlsanity.com/paper/2605.07278/) (2026)
- [A Control Theory of Predictability in Latent World Models](https://mlsanity.com/paper/2607.10362/) (2026)
- [On the Identifiability of Controlled World Models](https://mlsanity.com/paper/2607.22430/) (2026)
- [Why and How Auxiliary Tasks Improve JEPA Representations](https://mlsanity.com/paper/2509.12249/) (2025)
- [A Generalization Theory for JEPA-Based World Models](https://mlsanity.com/paper/2606.27014/) (2026)
- [Certified World Models: Predictability Across Configuration, Horizon, and Resolution](https://mlsanity.com/paper/2606.13092/) (2026)
- [ActSWM: Action-Sensitive World Models for Long-Horizon Planning in Open-World Games](https://mlsanity.com/paper/2607.26712/) (2026)
- [ProWorld: Progress-Aware Hyperbolic World Models for Long-Horizon Visual Goal Reaching](https://mlsanity.com/paper/2608.01926/) (2026)

## Abstract (arXiv)

We establish a finite-sample learning-to-control theory for geometrically supervised latent models of nonlinear deterministic systems. Geometric supervision is used only during training: simulator state, proprioception, or state estimates with independently validated metric and directional error bounds supply observable-state distances and tangent directions, while deployment remains observation- and action-conditioned. We introduce an encoder-only local--global metric hinge that enforces directional resolution and separated-state discrimination. Under regular observable-factor, coverage, finite-capacity approximation, and uniform C1,1 hypotheses, a computable one-sided regularization regime has a strong selection property: with high probability, every approximate empirical minimizer is simultaneously pointwise co-Lipschitz and uniformly approximately semiconjugate to the controlled dynamics. Approximation, sampling, and optimization errors remain explicit and separate. Norm-constrained tensor-product B-spline classes constructively realize the approximation hypotheses, and the interpolation exponent converting mean residual control into a uniform bound is sharp. A modular deterministic corollary transfers the learned certificates to trajectory, finite-horizon cost, learned-cost-head, and optimizer guarantees, while a validated finite-net result enables sharper model-specific certification. Controlled experiments isolate collapse and folding, quantify the analytic certificate's reserve, and demonstrate the control benefit of restored metric resolution. The principal contribution is a complete finite-sample implication from approximate empirical optimization to metric faithfulness, uniform controlled dynamics, and reliable planning for the same learned model.