# Operator-on-F complements value-equivalence: a planning-time diagnostic for latent world models

arXiv: https://arxiv.org/abs/2607.04464 · HTML: https://mlsanity.com/paper/2607.04464/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Latent dynamics models and model-based RL, one of 11 directions in World models.
- Builds on 10 of its 15 references that are in the map.
- First posted on arXiv 5 Jul 2026.

## Placed in

- World models / [Latent dynamics models and model-based RL](https://mlsanity.com/wm/latent-dynamics-models-and-model-based-rl/)
- Deep reinforcement learning methods / [Deep RL algorithms and representations](https://mlsanity.com/deeprl/deep-rl-algorithms-and-representations/)

## Builds on (most cited first, 10 of 10)

- [Understanding intermediate layers using linear classifier probes](https://mlsanity.com/paper/1610.01644/) (2016)
- [Mastering Diverse Domains through World Models](https://mlsanity.com/paper/2301.04104/) (2023)
- [Designing and Interpreting Probes with Control Tasks](https://mlsanity.com/paper/1909.03368/) (2019)
- [TD-MPC2: Scalable, Robust World Models for Continuous Control](https://mlsanity.com/paper/2310.16828/) (2023)
- [Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control](https://mlsanity.com/paper/1611.03537/) (2016)
- [The Value Equivalence Principle for Model-Based Reinforcement Learning](https://mlsanity.com/paper/2011.03506/) (2020)
- [The Predictron: End-To-End Learning and Planning](https://mlsanity.com/paper/1612.08810/) (2016)
- [Proper Value Equivalence](https://mlsanity.com/paper/2106.10316/) (2021)
- [Task-Oriented Koopman-Based Control with Contrastive Encoder](https://mlsanity.com/paper/2309.16077/) (2023)
- [Koopman Invariants as Drivers of Emergent Time-Series Clustering in Joint-Embedding Predictive Architectures](https://mlsanity.com/paper/2511.09783/) (2025)

## Built on by (most cited first, 2 of 2)

- [Diagnosing JEPA World Models with Action-Conditioned Predictive Consistency](https://mlsanity.com/paper/2608.12939/) (2026)
- [What a World Model Represents Is Three Questions](https://mlsanity.com/paper/2607.06640/) (2026)

## Abstract (arXiv)

World-model evaluation for model-based reinforcement learning typically asks whether the learned model predicts reward and value well, which can leave planning-relevant errors in the model's latent rollouts unmeasured. We introduce a complementary diagnostic, operator-on-F, that compares a model's k-step latent pushforward to the environment's on an observable subset F, using the model's own predictor. On a TD-MPC2 size sweep over cheetah-run, reward-prediction error stays within [0.028, 0.091] for every model size - only about 3x variation - so an unnormalized reward-fit check has narrow resolution to distinguish them; the (unnormalized) Bellman residual and reward error themselves have weak relationships with return (Spearman -0.10 and -0.30). Operator error spans 0.28 to 2.62 over the same sizes. At 317M the operator error is 2.62 - an order of magnitude above the 0.28-0.36 cluster - and the planning return collapses to 0.9, while reward-prediction error (0.091) is the highest of the five but stays within the same small [0.028, 0.091] range as the rest of the sweep. The rank correlation between operator error and return loss is -0.90 (anchor-bootstrap 95% CI [-0.90, -0.70] at n=5 sizes; leave-one-out removal of any single size leaves it at -0.80 or stronger). The operator also returns informative, architecture-discriminating estimates in a cross-architecture comparison between TD-MPC2 and a pure-SSL latent world model. The operator diagnostic complements value-equivalence rather than replacing it.