Predicting Closed-Loop Performance of Latent World Models: Offline Checkpoint Selection for MPC and Model-Based RL Under Non-Markovian Rewards in LunarLander

# Predicting Closed-Loop Performance of Latent World Models: Offline Checkpoint Selection for MPC and Model-Based RL Under Non-Markovian Rewards in LunarLander
Source: [https://arxiv.org/html/2607.01736](https://arxiv.org/html/2607.01736)
###### Abstract

We study how to predict the downstream closed\-loop performance of a learned latent world model from validation\-time diagnostics alone\. Choosing the right checkpoint from a world\-model training run is difficult: validation loss and multi\-step prediction RMSE keep improving long after closed\-loop performance has collapsed\. We present a suite of structural validation\-time diagnostics drawn from optimal\-control theory and apply them to Gymnasium’s LunarLander\-v3, which features shaped rewards\. We train an RSSM\[[5](https://arxiv.org/html/2607.01736#bib.bib4),[4](https://arxiv.org/html/2607.01736#bib.bib5)\]world model on it and treat per\-checkpoint CEM\-MPC return as the oracle for closed\-loop quality\. By evaluating4040metrics against this oracle, we find that the strongest single predictor is the Reward Observability Fraction \(ROF\), which measures the reward predictor’s dependence on the observable subspace\. We combine ROF with three structural regularizers into a single\-number offline checkpoint\-selection score, the Composite Reward Observability Fraction \(CROF\)\. The CROF\-selected world model trains a model\-based A2C policy that beats a fairly evaluated model\-free A2C baseline by∼24\.5\{\\sim\}24\.5return points while using∼65×\{\\sim\}65\\timesfewer real\-environment interactions, and the same world model also drives a strong zero\-shot CEM\-MPC policy\. Code and data:[https://github\.com/nsmoly/LunarLander\_RSSM](https://github.com/nsmoly/LunarLander_RSSM)\.

## 1Introduction

Model\-based reinforcement learning promises to reduce the number of real\-environment interactions needed to learn effective policies by building a predictive model of the environment—a*world model*—and using it for planning or policy optimization in imagination\. The Dreamer or Recurrent State Space Model \(RSSM\) architecture\[[5](https://arxiv.org/html/2607.01736#bib.bib4),[4](https://arxiv.org/html/2607.01736#bib.bib5)\]has become a standard backbone for world models, enabling both online planning via Model Predictive Control \(MPC\)\[[5](https://arxiv.org/html/2607.01736#bib.bib4)\]and policy learning with Reinforcement Learning \(RL\) through imagined rollouts\[[4](https://arxiv.org/html/2607.01736#bib.bib5),[6](https://arxiv.org/html/2607.01736#bib.bib6)\]\.

![Refer to caption](https://arxiv.org/html/2607.01736v1/figures/moonlander_mpc_1.jpg)Figure 1:LunarLander\-v3: four discrete actions \(NOP, left/main/right thruster\),88\-D observation, soft\-landing objective from a randomized initial state\.A fundamental challenge in model\-based RL is*objective mismatch*\[[8](https://arxiv.org/html/2607.01736#bib.bib8)\]: the training objective for the world model \(prediction accuracy\) does not necessarily align with the downstream objective \(closed\-loop performance\)\. A model with excellent one\-step or even multi\-step prediction accuracy may still be unsuitable for planning or policy optimization if its learned latent dynamics lack the structural properties needed for these purposes\.

In this work, we focus on the LunarLander\-v3 Gymnasium environment\. It has not been the subject of detailed RSSM/Dreamer\-style analysis in prior work and combines two useful properties: an88\-dimensional observation and44discrete actions, both small enough to admit direct structural analysis of the learned latent, and—the property that motivates this paper—a reward that is not fully recoverable from the current observation and action alone because of potential\-based shaping plus terminal events gated by simulator\-internal flags \(Section[4\.8](https://arxiv.org/html/2607.01736#S4.SS8)\)\. LunarLander thus serves as a tractable proxy for the broader class of environments with non\-Markovian or shaped reward, where standard one\-step prediction metrics turn out to be poor checkpoint\-selection criteria\. We develop structural diagnostics drawing on classical control theory \(controllability, observability, Kalman\[[7](https://arxiv.org/html/2607.01736#bib.bib1)\]\) and show experimentally that they identify world\-model checkpoints that produce substantially better downstream policies than standard model\-selection criteria, both for MPC and for model\-based actor–critic policy trained in imagination\.

Our contributions in this work are:

1. 1\.A suite of4040structural validation\-time metrics for world\-model evaluation, drawing on classical control theory: Jacobian\-based controllability/observability analysis \(fixed and time\-varying linearizations\) and three reward/observation subspace\-alignment scores\.
2. 2\.The Composite Reward Observability Fraction \(CROF\), a single\-number offline checkpoint\-selection score that combines reward/observation subspace\-alignment score with three structural scores \(controllability rank, observability rank, open\-loop observation error\)\.
3. 3\.Experimental evidence on LunarLander\-v3 that CROF tracks both zero\-shot CEM\-MPC returns and downstream A2C training quality, enabling offline selection of better world\-model checkpoints for closed\-loop use, while standard selection criteria \(validation loss, multi\-step RMSE, sensitivities\) pick overfit checkpoints with poor closed\-loop performance\.
4. 4\.A data\-efficiency comparison showing that the CROF\-selected world model trains a model\-based A2C policy that beats a fairly evaluated model\-free A2C baseline using∼65×\{\\sim\}65\\timesfewer real\-environment interactions\.

## 2Related Work

Our work draws on three areas: world models for control, the objective mismatch problem in model\-based RL, and classical control\-theoretic analysis of dynamical systems\.

### 2\.1World Models and Model\-Based RL

The RSSM family — PlaNet\[[5](https://arxiv.org/html/2607.01736#bib.bib4)\], Dreamer\[[4](https://arxiv.org/html/2607.01736#bib.bib5)\], and DreamerV3\[[6](https://arxiv.org/html/2607.01736#bib.bib6)\]— learns latent dynamics for planning and imagination\-based actor–critic training; PETS\[[2](https://arxiv.org/html/2607.01736#bib.bib7)\]pairs probabilistic ensemble dynamics with MPC for strong sample efficiency\. All of these methods train by maximizing prediction likelihood and evaluate by downstream task performance, without analyzing the structural properties of the learned dynamics that determine planning quality\.

### 2\.2Objective Mismatch

Lambert et al\.\[[8](https://arxiv.org/html/2607.01736#bib.bib8)\]identified the*objective mismatch*problem: optimizing a dynamics model for one\-step prediction accuracy does not guarantee good downstream closed\-loop performance, and proposed re\-weighting training objectives as an initial mitigation\. Our work proposes a complementary explanation of why this mismatch occurs: prediction accuracy \(even multi\-step\) captures statistical fit, while planning quality depends on the structural alignment between reward\-relevant, controllable, and observable subspaces — a geometric property that training losses do not measure\.

### 2\.3Task\-Relevant Representations

Several works learn representations that preserve task\-relevant information at training time: DeepMDP\[[3](https://arxiv.org/html/2607.01736#bib.bib9)\]via bisimulation\-inspired losses on reward and transition structure, SOLAR\[[14](https://arxiv.org/html/2607.01736#bib.bib11)\]by encouraging locally linear dynamics for LQR planning from images, and Agarwal et al\.\[[1](https://arxiv.org/html/2607.01736#bib.bib12)\]via contrastive embeddings of behavioral similarity\. These approaches modify the training objective\. Our work is complementary: we provide post\-hoc validation\-time diagnostics that quantify whether the learned structure supports closed\-loop planning, regardless of training procedure\.

### 2\.4Control\-Theoretic Analysis of Learned Systems

Controllability and observability originate with Kalman\[[7](https://arxiv.org/html/2607.01736#bib.bib1)\], who showed that the rank of the corresponding matrices determines whether a linear system can be fully steered and observed; Moore\[[9](https://arxiv.org/html/2607.01736#bib.bib2)\]extended this with balanced realization theory, identifying dimensions that are simultaneously controllable and observable\. Sussillo and Barak\[[13](https://arxiv.org/html/2607.01736#bib.bib10)\]applied local Jacobian and fixed\-point analysis to trained RNNs in neuroscience, revealing interpretable low\-dimensional dynamics in high\-dimensional recurrent networks\. We adapt these tools to the RSSM setting and combine them into scalar reward/observation alignment fractions \(RCF, OCF, ROF\) used as post\-hoc validation\-time diagnostics that predict closed\-loop performance without environment access\.

## 3Method

### 3\.1World Model Architecture

We use an RSSM\-inspired world model\[[4](https://arxiv.org/html/2607.01736#bib.bib5),[6](https://arxiv.org/html/2607.01736#bib.bib6)\]adapted for LunarLander\-v3, whose observation isp=8p\{=\}8\-dimensional \(6 continuous physics dimensions plus 2 binary leg\-contact indicators\) withK=4K\{=\}4discrete actions\. The RSSM core maintains a deterministic hidden state𝒉t∈ℝdh\\bm\{h\}\_\{t\}\\in\\mathbb\{R\}^\{d\_\{h\}\}\(dh=256d\_\{h\}\{=\}256, single\-layer GRU\) and a stochastic latent𝒛t∈ℝdz\\bm\{z\}\_\{t\}\\in\\mathbb\{R\}^\{d\_\{z\}\}\(dz=16d\_\{z\}\{=\}16\), giving a full latent state𝒔t=\[𝒉t;𝒛t\]∈ℝ272\\bm\{s\}\_\{t\}=\[\\bm\{h\}\_\{t\};\\,\\bm\{z\}\_\{t\}\]\\in\\mathbb\{R\}^\{272\}\. Actionsat∈\{0,1,2,3\}a\_\{t\}\\in\\\{0,1,2,3\\\}are one\-hot encoded as𝒖t∈ℝ4\\bm\{u\}\_\{t\}\\in\\mathbb\{R\}^\{4\}\. The RSSM transition \(prior\) is

𝒔t\+1=fθ​\(𝒔t,𝒖t\)=\[GRU⁡\(𝒉t,\[𝒛t;𝒖t\]\);𝝁prior​\(𝒉t\+1\)\],\\bm\{s\}\_\{t\+1\}=f\_\{\\theta\}\(\\bm\{s\}\_\{t\},\\bm\{u\}\_\{t\}\)\\;=\\;\\bigl\[\\operatorname\{GRU\}\(\\bm\{h\}\_\{t\},\\;\[\\bm\{z\}\_\{t\};\\bm\{u\}\_\{t\}\]\);\\;\\bm\{\\mu\}\_\{\\text\{prior\}\}\(\\bm\{h\}\_\{t\+1\}\)\\bigr\],\(1\)the posterior encoder isqθ​\(𝒛t∣𝒉t,𝒐t\)q\_\{\\theta\}\(\\bm\{z\}\_\{t\}\\mid\\bm\{h\}\_\{t\},\\bm\{o\}\_\{t\}\), and both prior and posterior are 2\-layer MLPs \(LayerNorm \+ SiLU\) predicting mean and log\-std of a diagonal Gaussian; the observation encoder is a 2\-layer MLP mapping𝒐t∈ℝ8\\bm\{o\}\_\{t\}\\in\\mathbb\{R\}^\{8\}to a feature vector concatenated with𝒉t\\bm\{h\}\_\{t\}for the posterior\.

##### Structured decoder with heterogeneous heads\.

Rather than treating all 8 observation dimensions identically, the decoder uses a shared 2\-layer MLP backbone \(LayerNorm \+ SiLU, inputdh\+dz=272d\_\{h\}\{\+\}d\_\{z\}\{=\}272\) feeding four heads: an MSE\-regression*physics*head \(ℝ6\\mathbb\{R\}^\{6\}, continuous states\), a BCE*contact*head \(ℝ2\\mathbb\{R\}^\{2\}logits, sigmoid at inference\), a BCE*done*head \(ℝ1\\mathbb\{R\}^\{1\}\), and a separate 3\-layer*reward*MLP \(ℝ272→256→256→1\\mathbb\{R\}^\{272\}\{\\to\}256\{\\to\}256\{\\to\}1\) that operates directly on\[𝒉;𝒛\]\[\\bm\{h\};\\bm\{z\}\]rather than the shared backbone, so the reward predictor develops its own representation without competing with reconstruction for backbone capacity\. The observation decoder is𝒐^t=gθ​\(𝒔t\)\\hat\{\\bm\{o\}\}\_\{t\}=g\_\{\\theta\}\(\\bm\{s\}\_\{t\}\)\(physics concatenated with sigmoid of contact logits\); the reward predictor isr^t=ρθ​\(𝒔t\)\\hat\{r\}\_\{t\}=\\rho\_\{\\theta\}\(\\bm\{s\}\_\{t\}\)\.

##### Training loss\.

The total loss is a weighted sum:

ℒ=wr​ℒrecon\+wρ​ℒrew\+wd​ℒdone\+wKL​ℒKL,\\mathcal\{L\}=w\_\{r\}\\mathcal\{L\}\_\{\\text\{recon\}\}\+w\_\{\\rho\}\\mathcal\{L\}\_\{\\text\{rew\}\}\+w\_\{d\}\\mathcal\{L\}\_\{\\text\{done\}\}\+w\_\{\\text\{KL\}\}\\,\\mathcal\{L\}\_\{\\text\{KL\}\},\(2\)with weightswr=1\.0w\_\{r\}\{=\}1\.0,wρ=1\.2w\_\{\\rho\}\{=\}1\.2,wd=0\.5w\_\{d\}\{=\}0\.5,wKL=1\.0w\_\{\\text\{KL\}\}\{=\}1\.0\(KL free\-bits floorβKL=0\.5\\beta\_\{\\text\{KL\}\}\{=\}0\.5\)\.ℒrecon\\mathcal\{L\}\_\{\\text\{recon\}\}combines MSE for physics and BCE for contacts;ℒrew\\mathcal\{L\}\_\{\\text\{rew\}\}is squared reward error;ℒdone\\mathcal\{L\}\_\{\\text\{done\}\}is done\-flag BCE;ℒKL=KL​\(qθ∥pθ\)\\mathcal\{L\}\_\{\\text\{KL\}\}=\\text\{KL\}\(q\_\{\\theta\}\\\|p\_\{\\theta\}\)is the posterior–prior KL divergence\.

### 3\.2MPC via Cross\-Entropy Method \(CEM\)

Since LunarLander\-v3 has a discrete action space \(4 actions: NOP, left thruster, main engine, right thruster\), standard gradient\-based planning does not apply\. We use the Cross\-Entropy Method \(CEM\)\[[11](https://arxiv.org/html/2607.01736#bib.bib3)\]adapted for discrete actions: at each real timestep, CEM iteratively refines a per\-step categorical distribution over a planning horizon ofH=25H\{=\}25steps\. Each iteration samplesN=384N\{=\}384candidate action sequences from the current distribution, rolls each one through the world\-model prior open\-loop while accumulating discounted reward∑k=1Hγk−1​r^t\+k\\sum\_\{k=1\}^\{H\}\\gamma^\{k\-1\}\\hat\{r\}\_\{t\+k\}\(γ=0\.97\\gamma\{=\}0\.97\), keeps the topE=48E\{=\}48elites, and updates the per\-step logits toward the elite action frequencies with smoothing factorα=0\.7\\alpha\{=\}0\.7\. AfterI=4I\{=\}4such iterations, the first action of the best sequence is executed and the procedure repeats at the next real timestep\. The planning horizonH=25H\{=\}25matches the rollout horizon used for metric evaluation, so the offline metrics measure exactly the regime CEM operates in\.

### 3\.3Actor\-Critic Policy via Latent Imagination

Following Dreamer\[[4](https://arxiv.org/html/2607.01736#bib.bib5)\], we train an actor–critic policy \(A2C\) entirely within the latent imagination of the world model\. The actor and critic are compact 3\-layer MLPs \(hidden dimension of 256\) that read the latent state\[𝒉;𝒛\]\[\\bm\{h\};\\bm\{z\}\]; the actor outputs a categorical distribution over the 4 actions, the critic predicts scalar value estimates\. Each training step samples a real observation–action sequence from the replay dataset, warms the RSSM posterior up for 5 steps on the real data, then imagines 15 further steps by rolling out the prior with actions sampled from the actor \(no observation correction\)\. Value targets areλ\\lambda\-returns\[[12](https://arxiv.org/html/2607.01736#bib.bib14)\]\(γ=0\.99\\gamma\{=\}0\.99,λ=0\.95\\lambda\{=\}0\.95\) computed from the world model’s reward head; the actor is updated by REINFORCE with advantage clipping, the critic by MSE, and entropy is decayed from0\.50\.5to0\.150\.15over training\. We train for10001000epochs at learning rate3×10−53\{\\times\}10^\{\-5\}and batch size256256\. Crucially, this stage uses*zero additional real\-environment interactions*: all learning happens in the learned world model on the same offline dataset used to train it\.

### 3\.4Metrics for World Model Quality

We define metrics in four categories; the full per\-metric list with empirical correlations against MPC return is consolidated in Table[1](https://arxiv.org/html/2607.01736#S4.T1)of Section[4\.5](https://arxiv.org/html/2607.01736#S4.SS5)\.

#### 3\.4\.1Validation Loss Decomposition

Five standard training diagnostics:val\_loss\(total weighted validation loss as in \([2](https://arxiv.org/html/2607.01736#S3.E2)\)\), its componentsval\_kl\(posterior–prior KL\) andval\_recon\(physics \+ contact reconstruction\), and the one\-step posterior RMSEspost\_obs\_rmseandpost\_rew\_rmse, which measure closed\-loop prediction quality with encoder corrections\.

#### 3\.4\.2Multi\-Step Open\-Loop Rollout

These metrics measure what CEM MPC experiences during planning: multi\-step prediction quality without encoder corrections\. After a 5\-step posterior warm\-up, the RSSM prior is rolled out for2525steps using ground\-truth actions, with squared errors masked over valid \(non\-padded, pre\-terminal\) positions and reduced by a square root\. We record nine quantities: observation RMSE at the first horizon step \(ol\_obs\_start\), averaged across the horizon \(ol\_obs\_avg\), the per\-rollout maximum \(ol\_obs\_max\), and at the last step \(ol\_obs\_end\); the same four for reward \(ol\_rew\_start,ol\_rew\_avg,ol\_rew\_max,ol\_rew\_end\); and the cumulative\-reward RMSEol\_cumrew\_errof∑k=1Hr^t\+k\\sum\_\{k=1\}^\{H\}\\hat\{r\}\_\{t\+k\}against ground truth\.

#### 3\.4\.3Jacobian\-Based Linear Analysis

We linearize the transition, decoder, and reward functions around sampled validation states and apply classical control\-theoretic analysis\[[7](https://arxiv.org/html/2607.01736#bib.bib1)\]\.

At a state𝒔=\[𝒉;𝒛\]\\bm\{s\}=\[\\bm\{h\};\\bm\{z\}\]with action𝒖\\bm\{u\}, the Jacobians are:

A\\displaystyle A=∂fθ∂𝒔\|𝒔,𝒖∈ℝn×n,\\displaystyle=\\frac\{\\partial f\_\{\\theta\}\}\{\\partial\\bm\{s\}\}\\biggr\|\_\{\\bm\{s\},\\bm\{u\}\}\\in\\mathbb\{R\}^\{n\\times n\},B\\displaystyle B=∂fθ∂𝒖\|𝒔,𝒖∈ℝn×K,\\displaystyle=\\frac\{\\partial f\_\{\\theta\}\}\{\\partial\\bm\{u\}\}\\biggr\|\_\{\\bm\{s\},\\bm\{u\}\}\\in\\mathbb\{R\}^\{n\\times K\},\(3\)C\\displaystyle C=∂gθ∂𝒔\|𝒔∈ℝp×n,\\displaystyle=\\frac\{\\partial g\_\{\\theta\}\}\{\\partial\\bm\{s\}\}\\biggr\|\_\{\\bm\{s\}\}\\in\\mathbb\{R\}^\{p\\times n\},R\\displaystyle R=∂ρθ∂𝒔\|𝒔∈ℝ1×n\.\\displaystyle=\\frac\{\\partial\\rho\_\{\\theta\}\}\{\\partial\\bm\{s\}\}\\biggr\|\_\{\\bm\{s\}\}\\in\\mathbb\{R\}^\{1\\times n\}\.\(4\)
##### Fixed vs\. time\-varying linearization\.

In*fixed*linearization \(jac\_prefix\), JacobiansA,B,C,RA,B,C,Rare computed once at a sampled state and the controllability/observability matrices use powersAkA^\{k\}:𝒞H=\[B,A​B,…,AH−1​B\]\\mathcal\{C\}\_\{H\}=\[B,\\,AB,\\,\\dots,\\,A^\{H\-1\}B\],𝒪H=\[C;C​A;…;C​AH−1\]\\mathcal\{O\}\_\{H\}=\[C;\\,CA;\\,\\dots;\\,CA^\{H\-1\}\]\. In*time\-varying*linearization \(jac\_dyn\_prefix\), fresh JacobiansAk,Bk,Ck,RkA\_\{k\},B\_\{k\},C\_\{k\},R\_\{k\}are computed at each step of an actual latent rollout through the prior and the matrices chain along that rollout\. Every Jacobian metric below is computed in both variants; Section[4\.6](https://arxiv.org/html/2607.01736#S4.SS6)compares them empirically\.

##### Spectral and rank summaries\.

FromAAand from𝒞H\\mathcal\{C\}\_\{H\},𝒪H\\mathcal\{O\}\_\{H\}we record six scalar summaries: the mean spectral radius ofAA\(jac\_spec\_radius\) and its per\-sample maximum \(jac\_spec\_radius\_max\); the effective controllability rankkck\_\{c\}\(jac\_ctrl\_rank\) and its singular\-value condition numberσ1​\(𝒞H\)/σkc​\(𝒞H\)\\sigma\_\{1\}\(\\mathcal\{C\}\_\{H\}\)/\\sigma\_\{k\_\{c\}\}\(\\mathcal\{C\}\_\{H\}\)\(jac\_ctrl\_cond\); and the analogousjac\_obs\_rankandjac\_obs\_condfor𝒪H\\mathcal\{O\}\_\{H\}\. Effective ranks use a threshold of10−310^\{\-3\}relative to the largest singular value\. The same six are recorded with thejac\_dyn\_prefix for time\-varying linearization \(jac\_dyn\_spec\_radius,jac\_dyn\_spec\_radius\_max,jac\_dyn\_ctrl\_rank,jac\_dyn\_ctrl\_cond,jac\_dyn\_obs\_rank,jac\_dyn\_obs\_cond\)\.

##### Subspace alignment metrics\.

SVD the controllability and observability matrices as𝒞H=Uc​Σc​Vc⊤\\mathcal\{C\}\_\{H\}=U\_\{c\}\\Sigma\_\{c\}V\_\{c\}^\{\\top\}and𝒪H=Uo​Σo​Vo⊤\\mathcal\{O\}\_\{H\}=U\_\{o\}\\Sigma\_\{o\}V\_\{o\}^\{\\top\}\. LetUc\(k\)∈ℝn×kcU\_\{c\}^\{\(k\)\}\\in\\mathbb\{R\}^\{n\\times k\_\{c\}\}andVo\(k\)∈ℝn×koV\_\{o\}^\{\(k\)\}\\in\\mathbb\{R\}^\{n\\times k\_\{o\}\}be the controllable and observable subspace bases,𝒓=R⊤∈ℝn\\bm\{r\}=R^\{\\top\}\\in\\mathbb\{R\}^\{n\}the reward gradient,∥⋅∥F\\lVert\\cdot\\rVert\_\{F\}the Frobenius norm, and∥⋅∥\\lVert\\cdot\\rVertthe Euclidean norm\. The three subspace\-alignment scores are:

- •Reward Controllability Fraction \(RCF\): fraction of the reward gradient’s energy in the controllable subspace\. RCF=∥Uc\(k\)⊤​𝒓∥2∥𝒓∥2∈\[0,1\]\.\\text\{RCF\}\\;=\\;\\frac\{\\lVert U\_\{c\}^\{\(k\)\\top\}\\bm\{r\}\\rVert^\{2\}\}\{\\lVert\\bm\{r\}\\rVert^\{2\}\}\\;\\in\\;\[0,1\]\.\(5\)
- •Observation Controllability Fraction \(OCF\): fraction of the decoder’s sensitivity in the controllable subspace\. OCF=∥C​Uc\(k\)∥F2∥C∥F2∈\[0,1\]\.\\text\{OCF\}\\;=\\;\\frac\{\\lVert C\\,U\_\{c\}^\{\(k\)\}\\rVert\_\{F\}^\{2\}\}\{\\lVert C\\rVert\_\{F\}^\{2\}\}\\;\\in\\;\[0,1\]\.\(6\)
- •Reward Observability Fraction \(ROF\): fraction of the reward gradient’s energy in the observable subspace\. ROF=∥Vo\(k\)⊤​𝒓∥2∥𝒓∥2∈\[0,1\]\.\\text\{ROF\}\\;=\\;\\frac\{\\lVert V\_\{o\}^\{\(k\)\\top\}\\bm\{r\}\\rVert^\{2\}\}\{\\lVert\\bm\{r\}\\rVert^\{2\}\}\\;\\in\\;\[0,1\]\.\(7\)

Each score is recorded for both linearizations:jac\_rcf,jac\_ocf,jac\_rof\(fixed\) andjac\_dyn\_rcf,jac\_dyn\_ocf,jac\_dyn\_rof\(time\-varying\)\.

##### Connection to control theory\.

Controllability and observability are classical prerequisites for optimal control\[[7](https://arxiv.org/html/2607.01736#bib.bib1)\]and carry over directly to MPC and latent\-imagination RL, which both roll out the world model through its prior\. ROF is the key metric in this regime: it measures the fraction of the reward gradient that lies in the observable subspace, i\.e\. in the directions that would normally be corrected by the encoder but drift uncorrected during open\-loop imagination \(see Section[4\.7](https://arxiv.org/html/2607.01736#S4.SS7)for the full mechanism\)\.

##### Curated state sampling\.

The ROF score is additionally computed twice per linearization: on “good” \(return≥\+100\\geq\{\+\}100\) and “bad” \(return≤−100\\leq\{\-\}100\) validation sequences, givingjac\_rof\(good, fixed\) andjac\_rof\_bad\(bad, fixed\), and analogouslyjac\_dyn\_rofandjac\_dyn\_rof\_badfor time\-varying\. Splitting the classes avoids the seed\-dependent batch\-composition variance produced by uniform mixing\. The headline state\-aware metric is the convex combination

jac\_rof\_combined=α⋅jac\_rof\+\(1−α\)⋅jac\_rof\_bad,\\texttt\{jac\\\_rof\\\_combined\}\\;=\\;\\alpha\\cdot\\texttt\{jac\\\_rof\}\\;\+\\;\(1\-\\alpha\)\\cdot\\texttt\{jac\\\_rof\\\_bad\},\(8\)withα=0\.5\\alpha\{=\}0\.5throughout \(a sweep confirms a broad optimum nearα∈\[0\.3,0\.5\]\\alpha\{\\in\}\[0\.3,0\.5\]\)\.

#### 3\.4\.4Empirical Sensitivity Metrics

These use finite\-difference estimation rather than Jacobians:emp\_Cis the mean pairwise distance between next states produced by different actions \(action sensitivity\);emp\_Ois the transition response to perturbations in𝒛\\bm\{z\}\(state\-perturbation sensitivity\); andemp\_Lis the*maximum*over sampled state pairs of∥fθ​\(𝒔1,𝒖\)−fθ​\(𝒔2,𝒖\)∥/∥𝒔1−𝒔2∥\\lVert f\_\{\\theta\}\(\\bm\{s\}\_\{1\},\\bm\{u\}\)\-f\_\{\\theta\}\(\\bm\{s\}\_\{2\},\\bm\{u\}\)\\rVert/\\lVert\\bm\{s\}\_\{1\}\-\\bm\{s\}\_\{2\}\\rVert\(the max captures localized blow\-ups that the mean would dilute\)\. Together with the two CROF composites defined in Section[3\.5](https://arxiv.org/html/2607.01736#S3.SS5), these complete the suite at5\+9\+21\+3\+2=405\+9\+21\+3\+2=40metrics per checkpoint\.

### 3\.5Composite Reward Observability Fraction \(CROF\)

jac\_rof\_combinedis the strongest single predictor of MPC performance, but at undertrained checkpoints the latent is barely organized and ROF can be artificially small\. We therefore introduce theComposite Reward Observability Fraction \(CROF\), combining ROF with three structural regularizers\. We report two variants that share the same four components but use different relative weights:

CROF\-A=ROF~\+1\.0⋅\(1−kc~\)\+1\.0⋅\(1−ko~\)\+1\.0⋅e~obs,\\displaystyle=\\widetilde\{\\text\{ROF\}\}\+1\.0\\cdot\(1\-\\widetilde\{k\_\{c\}\}\)\+1\.0\\cdot\(1\-\\widetilde\{k\_\{o\}\}\)\+1\.0\\cdot\\widetilde\{e\}\_\{\\text\{obs\}\},\(9\)CROF\-B=ROF~\+0\.5⋅\(1−kc~\)\+0\.5⋅\(1−ko~\)\+0\.5⋅e~obs,\\displaystyle=\\widetilde\{\\text\{ROF\}\}\+0\.5\\cdot\(1\-\\widetilde\{k\_\{c\}\}\)\+0\.5\\cdot\(1\-\\widetilde\{k\_\{o\}\}\)\+0\.5\\cdot\\widetilde\{e\}\_\{\\text\{obs\}\},\(10\)where\(⋅\)~\\widetilde\{\(\\cdot\)\}denotes min–max normalization across all checkpoints,ROF=jac\_rof\_combined\\text\{ROF\}=\\texttt\{jac\\\_rof\\\_combined\}from \([8](https://arxiv.org/html/2607.01736#S3.E8)\),kc=jac\_ctrl\_rankk\_\{c\}=\\texttt\{jac\\\_ctrl\\\_rank\},ko=jac\_obs\_rankk\_\{o\}=\\texttt\{jac\\\_obs\\\_rank\}, andeobs=ol\_obs\_avge\_\{\\text\{obs\}\}=\\texttt\{ol\\\_obs\\\_avg\}is the open\-loop observation RMSE averaged across all rollout steps\.Lower CROF==better checkpoint; all four normalized terms are oriented so that smaller is better\.

##### Rationale\.

ROF is the primary signal; the three regularizers protect against degenerate corners of it:\(1−kc\)\(1\{\-\}k\_\{c\}\)penalizes low controllability rank,\(1−ko\)\(1\{\-\}k\_\{o\}\)penalizes low observability rank, andeobse\_\{\\text\{obs\}\}penalizes high open\-loop observation RMSE \(a sanity check on the dynamics\)\. CROF\-A and CROF\-B differ only in how heavily these regularizers are weighed against the dominant ROF term; we report both as a transparent ablation rather than picking a single recipe a priori\.

## 4Experiments

### 4\.1Environment and Data

We use Gymnasium’s LunarLander\-v3 \(Section[3\.1](https://arxiv.org/html/2607.01736#S3.SS1)\)\. The reward decomposes into per\-step shaping plus terminal events; we give the explicit formula and analyze its observability in Section[4\.8](https://arxiv.org/html/2607.01736#S4.SS8)\.

We collected872872human\-piloted episodes \(180,916180\{,\}916environment steps;158,685158\{,\}685train \+22,23122\{,\}231validation\), split750/122750/122train/val \(32,03832\{,\}038/4,5004\{,\}500valid sequence positions at length3030, stride55\)\. Two pilots took turns to introduce stylistic diversity, and the mix deliberately covers clean landings, near\-misses, and crashes so the world model sees both reward\-positive and reward\-negative regimes\. The dataset is collected once and reused for all model\-based training and metric evaluation\.

### 4\.2World Model Training

The RSSM world model \(WM\) is trained for500500epochs with learning rate10−410^\{\-4\}\(AdamW\), batch size6464, sequence length3030at stride55, KL free\-bits floorβKL=0\.5\\beta\_\{\\text\{KL\}\}\{=\}0\.5, and loss weights as in \([2](https://arxiv.org/html/2607.01736#S3.E2)\)\. Checkpoints are saved every55epochs, yielding100100for evaluation\. All random seeds in this paper \(dataloader shuffling, MPC and A2C evaluation episodes, metric sampling, model initialization\) are fixed to1234512345so every pair of reported numbers uses identical random draws\.

### 4\.3MPC Evaluation and Metrics

Each of the 100 world model checkpoints \(epochs 5, 10, …, 500\) is evaluated using CEM\-MPC \(Section[3\.2](https://arxiv.org/html/2607.01736#S3.SS2)\) with deterministic seeding for 20 episodes per checkpoint\. We report the mean return across these 20 episodes\. CEM parameters: horizonH=25H\{=\}25, populationN=384N\{=\}384, elitesE=48E\{=\}48, iterationsI=4I\{=\}4, smoothingα=0\.7\\alpha\{=\}0\.7, planning discountγ=0\.97\\gamma\{=\}0\.97\.

All metrics from Section[3\.4](https://arxiv.org/html/2607.01736#S3.SS4)are computed for each of the 100 checkpoints on the validation set\. Per\-class ROF metrics additionally use the curated good \(R≥\+100R\{\\geq\}\{\+\}100,2,7132\{,\}713stride\-11positions\) and bad \(R≤−100R\{\\leq\}\{\-\}100,1,0271\{,\}027positions\) subsets\. Jacobian analysis usesNJ=128N\_\{J\}\{=\}128sampled latent states per checkpoint \(per state\-class for the curated metrics\), each collected after a 5\-step posterior warm\-up, with a 25\-step rollout horizon matching the CEM planning horizon\.

### 4\.4MPC Performance Over Trained World Model Checkpoints

Figure[2](https://arxiv.org/html/2607.01736#S4.F2)shows CEM\-MPC performance across all 100 world model checkpoints\. Per\-checkpoint mean returns are highly variable \(std≈80\\approx 80–120120across2020episodes, dominated by LunarLander’s stochastic initial conditions\), so we use the 7\-point moving\-average \(MA\-7\) curve —140140pooled episodes per smoothed point — as the selection oracle\. Three regimes emerge:

- •Early\-training / improving\(≲200\{\\lesssim\}200\): the raw curve has occasional spikes \(e\.g\. epoch6060at\+173\.5\+173\.5\) but the smoothed curve stays below\+100\+100throughout, so these are noisy outliers rather than usable checkpoints\.
- •Smoothed peak: the MA\-7 mean peaks at epoch310310\(\+153\.0\+153\.0\) over a broad plateau spanning epochs∼260\{\\sim\}260–320320\. Selecting the right checkpoint inside this plateau is what CROF is designed to do offline\.
- •Late\-training collapse\(≳380\{\\gtrsim\}380\): the smoothed return drops sharply \(e\.g\. epoch405405at−55\.4\-55\.4\)\. The collapse is invisible to standard training metrics \(validation loss, multi\-step RMSE\), which keep improving monotonically\.

![Refer to caption](https://arxiv.org/html/2607.01736v1/figures/mpc_performance.png)Figure 2:CEM\-MPC performance across500500epochs of world\-model training \(2020test episodes per checkpoint;140140pooled per MA\-7 point\)\. Bold/faint blue: MA\-7\-smoothed and raw mean\. Green: MA\-7\-smoothed median\. Light band: MA\-7 min–max envelope \(a few extreme points are clipped\)\. The smoothed mean peaks near epoch310310and collapses after epoch380380\.
### 4\.5Metric Correlation Analysis

We compute Pearsonrr, Spearmanρs\\rho\_\{s\}, and quadraticRq2R^\{2\}\_\{q\}for each metric against the smoothed MPC mean return \(7\-point moving average\)\. Table[1](https://arxiv.org/html/2607.01736#S4.T1)consolidates the4040\-metric suite from Section[3\.4](https://arxiv.org/html/2607.01736#S3.SS4)into one ranked reference: each row lists what the metric measures and its three correlation scores, with strongly correlated rows in bold\. Throughout the paper we use the thresholds*strong*:\|ρs\|≥0\.30\|\\rho\_\{s\}\|\\geq 0\.30,*moderate*:0\.10≤\|ρs\|<0\.300\.10\\leq\|\\rho\_\{s\}\|<0\.30, and*weak*:\|ρs\|<0\.10\|\\rho\_\{s\}\|<0\.10\. Figure[6](https://arxiv.org/html/2607.01736#A1.F6)in Appendix[A](https://arxiv.org/html/2607.01736#A1)shows the same Spearman correlations as a bar chart across all4040metrics, and Figure[7](https://arxiv.org/html/2607.01736#A1.F7)in the same appendix shows each metric’s full curve over training \(only the ROF family traces MPC’s rise–peak–collapse U\-shape\)\. ROF based metrics have the strongest correlation with the MPC returns, with Spearmanρs\\rho\_\{s\}in the range−0\.71≤ρs≤−0\.395\-0\.71\\leq\\rho\_\{s\}\\leq\-0\.395\. Some metrics like the validation loss have weak correlation with MPC returns, see Table[1](https://arxiv.org/html/2607.01736#S4.T1)\.

MetricWhat it measuresPearsonrrSpearmanρs\\rho\_\{s\}Rq2R^\{2\}\_\{q\}*Strong predictors \(\|ρs\|≥0\.30\|\\rho\_\{s\}\|\\geq 0\.30, bold\)*jac\_rof\_combined0\.5⋅jac\_rof\+0\.5⋅jac\_rof\_bad0\.5\\cdot\\texttt\{jac\\\_rof\}\+0\.5\\cdot\\texttt\{jac\\\_rof\\\_bad\}, state\-aware ROF−0\.683\\mathbf\{\-0\.683\}−0\.710\\mathbf\{\-0\.710\}0\.520\\mathbf\{0\.520\}jac\_rof\_badFixed\-Jacobian ROF on bad \(low\-return\) sequences−0\.656\\mathbf\{\-0\.656\}−0\.692\\mathbf\{\-0\.692\}0\.434\\mathbf\{0\.434\}jac\_rofFixed\-Jacobian ROF on good \(high\-return\) sequences−0\.558\\mathbf\{\-0\.558\}−0\.648\\mathbf\{\-0\.648\}0\.479\\mathbf\{0\.479\}CROF\-BComposite: ROF\+0\.5×\+\\,0\.5\{\\times\}rank/OL\-err regularizers−0\.495\\mathbf\{\-0\.495\}−0\.568\\mathbf\{\-0\.568\}0\.421\\mathbf\{0\.421\}jac\_dyn\_rofTime\-varying ROF on good sequences−0\.467\\mathbf\{\-0\.467\}−0\.477\\mathbf\{\-0\.477\}0\.229\\mathbf\{0\.229\}CROF\-AComposite: ROF\+1\.0×\+\\,1\.0\{\\times\}rank/OL\-err regularizers−0\.375\\mathbf\{\-0\.375\}−0\.443\\mathbf\{\-0\.443\}0\.243\\mathbf\{0\.243\}jac\_dyn\_rof\_badTime\-varying ROF on bad sequences−0\.277\\mathbf\{\-0\.277\}−0\.395\\mathbf\{\-0\.395\}0\.182\\mathbf\{0\.182\}ol\_rew\_endOpen\-loop reward RMSE at horizon step2525\+0\.272\\mathbf\{\+0\.272\}\+0\.301\\mathbf\{\+0\.301\}0\.101\\mathbf\{0\.101\}val\_klKL component of validation loss\+0\.206\\mathbf\{\+0\.206\}\+0\.300\\mathbf\{\+0\.300\}0\.064\\mathbf\{0\.064\}*Moderate predictors \(0\.10≤\|ρs\|<0\.300\.10\\leq\|\\rho\_\{s\}\|<0\.30\)*jac\_dyn\_obs\_condCondition number of time\-varying observability matrix\+0\.243\+0\.243\+0\.273\+0\.2730\.0740\.074val\_reconReconstruction component of validation loss−0\.082\-0\.082−0\.184\-0\.1840\.0180\.018jac\_spec\_radiusSpectral radius of fixed\-Jacobian dynamics−0\.157\-0\.157−0\.172\-0\.1720\.0550\.055emp\_LMax\-Lipschitz offθf\_\{\\theta\}over state pairs\+0\.327\+0\.327\+0\.154\+0\.1540\.1100\.110emp\_OState\-perturbation sensitivity \(finite\-difference\)\+0\.104\+0\.104\+0\.130\+0\.1300\.0510\.051ol\_rew\_startOpen\-loop reward RMSE at horizon step11−0\.156\-0\.156−0\.128\-0\.1280\.0860\.086*Representative weak / near\-zero baselines \(\|ρs\|<0\.10\|\\rho\_\{s\}\|<0\.10\)*val\_lossTotal weighted validation loss\+0\.029\+0\.029\+0\.090\+0\.0900\.0150\.015post\_obs\_rmseOne\-step posterior observation RMSE−0\.084\-0\.084\+0\.012\+0\.0120\.0100\.010post\_rew\_rmseOne\-step posterior reward RMSE\+0\.028\+0\.028\+0\.085\+0\.0850\.0180\.018ol\_obs\_avgOpen\-loop observation RMSE averaged over horizon−0\.087\-0\.087\+0\.017\+0\.0170\.0110\.011ol\_cumrew\_errOpen\-loop cumulative\-reward RMSE\+0\.002\+0\.002\+0\.107\+0\.1070\.0620\.062jac\_ctrl\_rankEffective controllability rank \(fixed Jacobian\)\+0\.113\+0\.113\+0\.015\+0\.0150\.0130\.013emp\_CAction sensitivity \(finite\-difference\)\+0\.069\+0\.069−0\.093\-0\.0930\.0290\.029Table 1:Representative metrics from the4040\-metric suite \(Section[3\.4](https://arxiv.org/html/2607.01736#S3.SS4)\), ranked by Spearman\|ρs\|\|\\rho\_\{s\}\|against smoothed MPC mean return \(MA\-7,N=100N\{=\}100checkpoints\)\. To save space we only list the most informative metrics by correlation and a few additional metrics included by relevance \(standard training/prediction baselines, control\-theoretic ranks, finite\-difference sensitivities\); the remaining∼\\sim1818metrics not listed all sit inside the weak band\. Top block \(bold\): the99strongly correlated metrics — the ROF/CROF family dominates, joined byol\_rew\_endandval\_kl\. Middle block:66moderately correlated metrics\. Bottom block:77near\-zero baselines, where all standard training and one\-step\-prediction metrics fall\. Equations:jac\_rof\_combined\([8](https://arxiv.org/html/2607.01736#S3.E8)\), ROF \([7](https://arxiv.org/html/2607.01736#S3.E7)\), CROF\-A/B \([9](https://arxiv.org/html/2607.01736#S3.E9)\)/\([10](https://arxiv.org/html/2607.01736#S3.E10)\)\.
### 4\.6Fixed vs\. Time\-Varying Jacobian Analysis

Contrary to our initial expectation, the*fixed*linearization is the stronger MPC predictor\. The clearest case is ROF:jac\_rof\(fixed, good states\) reachesρs=−0\.65\\rho\_\{s\}\{=\}\{\-\}0\.65,Rq2=0\.48R^\{2\}\_\{q\}\{=\}0\.48vs\.ρs=−0\.48\\rho\_\{s\}\{=\}\{\-\}0\.48,Rq2=0\.23R^\{2\}\_\{q\}\{=\}0\.23forjac\_dyn\_rof, and the gap on the bad\-state subscore widens toρs=−0\.69\\rho\_\{s\}\{=\}\{\-\}0\.69vs\.−0\.40\{\-\}0\.40\(numbers from Table[1](https://arxiv.org/html/2607.01736#S4.T1)\)\. The remaining structural sub\-metrics \(spec\_radius, ctrl/obs\_rank, RCF, OCF\) are weakly correlated in both variants and neither wins consistently\. We believe the fixed variant wins because chainingH−1H\{\-\}1Jacobians along an open\-loop rollout compounds linearization error while CEM\-MPC re\-plans each step, so a single Jacobian at the current warm\-up\-grounded state is closer to the regime CEM actually operates in\. We therefore drop time\-varying metrics from the CROF formulation; they remain available in the released code\.

![Refer to caption](https://arxiv.org/html/2607.01736v1/figures/rof_analysis.png)Figure 3:ROF analysis\. Top\-left: smoothed MPC return \(blue\) vs\. invertedjac\_rof\_combined\(red\), with vertical lines at the smoothed MPC peak \(epoch310310\) and the smoothedjac\_rof\_combinedminimum \(epoch265265\)\. Top\-right: per\-checkpoint scatter with quadratic fit, exhibiting the characteristic U\-shape \(smoothed Spearmanρs=−0\.71\\rho\_\{s\}\{=\}\-0\.71,Rq2=0\.52R^\{2\}\_\{q\}\{=\}0\.52\)\. Bottom\-left: checkpoint selection comparison — the MA\-7\-smoothedjac\_rof\_combinedminimum and the MA\-7\-smoothed CROF\-A and CROF\-B picks all sit inside or adjacent to the high\-MPC plateau \(raw minima in Table[2](https://arxiv.org/html/2607.01736#S4.T2)\)\. Bottom\-right: subspace alignment metrics \(RCF, OCF, ROF\) vs\. epoch\.
### 4\.7ROF as a Predictor of MPC Performance

jac\_rof\_combinedis the strongest single predictor in our suite:ρs=−0\.71\\rho\_\{s\}=\-0\.71\(N=100N\{=\}100,p≲10−15p\\lesssim 10^\{\-15\},Rq2=0\.52R^\{2\}\_\{q\}=0\.52; visualized in Figure[3](https://arxiv.org/html/2607.01736#S4.F3)\)\. Both per\-class subscores are strong on their own \(ρs\\rho\_\{s\}of−0\.65\-0\.65for good states,−0\.69\-0\.69for bad states\), so the combined score rests on two largely independent pieces of evidence rather than a single regression artefact\.

##### Analysis\.

ROF measures how much of the reward gradient lies in the observable subspace\. The posterior anchors that subspace during training using real observations, but CEM\-MPC and latent\-imagination A2C roll the prior in imagination forH=25H\{=\}25steps with no encoder corrections, so the observable directions drift uncorrected\. A reward head that reads predominantly from those directions \(high ROF\) therefore degrades along the rollout, while a head that reads from the complementary subspace \(low ROF\) remains accurate\. Section[4\.8](https://arxiv.org/html/2607.01736#S4.SS8)confirms that LunarLander’s reward sits exactly in this regime:∼71%\{\\sim\}71\\%of per\-step reward variance is unrecoverable from\(ot,at\)\(o\_\{t\},a\_\{t\}\)alone, so a well\-trained world model must encode that part outside the observation\-decoder column space\.

### 4\.8Reward Predictability from Observations

LunarLander’s per\-step reward, taken from the Gymnasium reference implementation,111[https://github\.com/Farama\-Foundation/Gymnasium/blob/main/gymnasium/envs/box2d/lunar\_lander\.py](https://github.com/Farama-Foundation/Gymnasium/blob/main/gymnasium/envs/box2d/lunar_lander.py)has the form

rt=ϕ​\(ot\+1\)−ϕ​\(ot\)⏟PBRS shaping−cmain​1​\[at=main\]−cside​1​\[at∈\{L,R\}\]\+rtterm,r\_\{t\}\\;=\\;\\underbrace\{\\phi\(o\_\{t\+1\}\)\-\\phi\(o\_\{t\}\)\}\_\{\\text\{PBRS shaping\}\}\\;\-\\;c\_\{\\text\{main\}\}\\,\\mathbb\{1\}\[a\_\{t\}\{=\}\\text\{main\}\]\\;\-\\;c\_\{\\text\{side\}\}\\,\\mathbb\{1\}\[a\_\{t\}\{\\in\}\\\{\\text\{L,R\}\\\}\]\\;\+\\;r^\{\\text\{term\}\}\_\{t\},\(11\)with shaping potentialϕ​\(o\)=−100​\(‖p‖\+‖v‖\)−100​\|θ\|\+10​\(leg1\+leg2\)\\phi\(o\)=\-100\(\\\|p\\\|\+\\\|v\\\|\)\-100\|\\theta\|\+10\(\\text\{leg\}\_\{1\}\+\\text\{leg\}\_\{2\}\)defined on the visible state,\(cmain,cside\)=\(0\.30,0\.03\)\(c\_\{\\text\{main\}\},c\_\{\\text\{side\}\}\)=\(0\.30,0\.03\), andrtterm∈\{−100,\+100,0\}r^\{\\text\{term\}\}\_\{t\}\\in\\\{\-100,\+100,0\\\}awarded only at episode end and gated by Box2D\-internal flags \(game\_over,lander\.awake\) absent fromoto\_\{t\}\. Two pieces of this expression—theϕ​\(ot\+1\)\\phi\(o\_\{t\+1\}\)term andrttermr^\{\\text\{term\}\}\_\{t\}—are structurally not recoverable from\(ot,at\)\(o\_\{t\},a\_\{t\}\), which is what we measure below\.

To measure how much ofrtr\_\{t\}is recoverable from observations alone, we trained MLP regressors \(two hidden layers of256256units, ReLU, AdamW, MSE loss\) on the same dataset, in four input configurations and computedR2R^\{2\}score for each that measures the percentage of the data the model can explain from its input\. We got the following results:R2R^\{2\}:0\.2900\.290for\(ot,at\)\(o\_\{t\},a\_\{t\}\)input alone,0\.6700\.670when terminal events are filtered out \(steps wheredone=1\\text\{done\}\{=\}1which account<1%\{<\}1\\%of data\),0\.5950\.595whenot\+1o\_\{t\+1\}is added, and0\.971\\mathbf\{0\.971\}when both terminals are filtered andot\+1o\_\{t\+1\}is added\.

Reading the decomposition:\(ot,at\)\(o\_\{t\},a\_\{t\}\)alone leaves∼71%\\sim 71\\%of per\-step reward variance unexplainable; addingot\+1o\_\{t\+1\}and filtering terminals recovers more reward\. A sequence\-model latent can track the missing context from the trajectory history, so the reward head must read from directions outside the observation\-decoder column space\. ROF measures the fraction of the reward gradient that lies*inside*that column space, so lower ROF marks the checkpoints where the reward head has moved its computation to where it belongs\.

##### Reacher: a Markovian\-reward contrast\.

As a sanity check on the structural argument, we re\-ran the same MLP regression and the same Jacobian\-based diagnostic suite on a Gymnasium Reacher world model trained with an identical RSSM setup\. Predictingrtr\_\{t\}from\(ot,at\)\(o\_\{t\},a\_\{t\}\)alone gives MLP validationR2=0\.9998R^\{2\}=0\.9998: Reacher’s reward is fully Markovian in the observation–action pair\. Correspondingly,jac\_rof\_combinedis essentially uninformative about closed\-loop performance — Spearmanρs=\+0\.143\\rho\_\{s\}=\{\+\}0\.143against smoothed MPC return \(Rq2=0\.265R^\{2\}\_\{q\}=0\.265\), opposite in sign to LunarLander’s−0\.71\-0\.71, and MPC return rises nearly monotonically over training rather than tracing the LunarLander U\-shape \(Figure[4](https://arxiv.org/html/2607.01736#S4.F4)\)\. On this task there is no checkpoint\-selection problem for ROF to solve: standard one\-step prediction metrics work fine\.

![Refer to caption](https://arxiv.org/html/2607.01736v1/figures/reacher_rof_topleft.png)Figure 4:Reacher contrast\. Smoothed MPC return \(blue\) and invertedjac\_rof\_combined\(red\) over training on a Gymnasium Reacher world model trained with the identical RSSM setup\. Both curves rise nearly monotonically; there is no U\-shape and no late\-training collapse, and the rank correlation between the two \(ρs=\+0\.143\\rho\_\{s\}=\{\+\}0\.143,Rq2=0\.265R^\{2\}\_\{q\}=0\.265\) is weak and of the opposite sign to LunarLander\.
##### When ROF carries signal\.

The structural condition for ROF to be informative appears to be that the reward is not Markovian in\(ot,at\)\(o\_\{t\},a\_\{t\}\), so a well\-trained world model has to encode reward\-relevant quantities in latent directions outside the observation\-decoder column space\. This class includes tasks with PBRS shaping\[[10](https://arxiv.org/html/2607.01736#bib.bib15)\], tasks whose terminal or sparse rewards depend on hidden simulator state, and partially observed control tasks more generally\. LunarLander is a representative example of such setups\. We therefore present CROF as a metric targeting this practically common structural condition rather than as a domain\-agnostic rule\.

### 4\.9Checkpoint Selection via CROF and Structural Insights

Table[2](https://arxiv.org/html/2607.01736#S4.T2)compares checkpoint\-selection criteria: the smoothed\-MPC oracle, raw and smoothed\-min picks of the structural metrics, and representative weak baselines; Figure[5](https://arxiv.org/html/2607.01736#S4.F5)plots smoothed CROF\-A and CROF\-B against MPC return with the same picks marked\.

Selection ruleEpoch selectedMPC return*Oracle baseline*Best smoothed MPC \(oracle, MA\-7\)310\+153\.0\+153\.0*Structural / proposed metrics*Min CROF\-A \(raw\)280\+162\.8\+162\.8Min CROF\-A \(smoothed MA\-7\)315\+148\.1\+148\.1Min CROF\-B \(raw\)280\+162\.8\+162\.8Min CROF\-B \(smoothed MA\-7\)270\+117\.5\+117\.5Min jac\_rof\_combined \(raw\)250\+115\.8\+115\.8Min jac\_rof\_combined \(smoothed MA\-7\)265\+156\.5\+156\.5*Standard / weak baselines*Min val\_loss470\+79\.3\+79\.3Min post\_rew\_rmse470\+79\.3\+79\.3Min ol\_obs\_end495\+55\.9\+55\.9Min ol\_cumrew\_err460\+2\.9\+2\.9Max empCC485\+15\.6\+15\.6Max jac\_ctrl\_rank445\+115\.8\+115\.8Table 2:Checkpoint selection comparison\. The smoothed \(MA\-7\) MPC oracle peaks at epoch310310with a plateau over epochs260260–320320; both CROF variants’ raw\-minimum picks \(epoch280280\) land inside it\. The pure\-ROF baseline \(jac\_rof\_combined\) picks epoch250250raw /265265smoothed — only the smoothed pick is in the plateau, motivating the CROF auxiliary terms\. All standard criteria pick deeply overfit checkpoints \(epoch≥460\\text\{epoch\}\\geq 460\) with MPC returns of\+3\+3to\+79\+79\.ROF and CROF are also the only metrics whose informativeness peaks in the regime one would actually select from: a phase\-wise breakdown showsjac\_rof\_combinedreaches Pearsonr=−0\.65r\{=\}\{\-\}0\.65in mid\-training \(epochs 100–300\) versus−0\.20\-0\.20early and−0\.32\-0\.32late\.

Two observations from the Jacobian analysis tell the broader story\. First, controllability is low\-dimensional:kc≈33k\_\{c\}\\approx 33–4747out ofn=272n\{=\}272latent dimensions \(∼12\{\\sim\}12–17%17\\%\), with the GRU’s gating preserving historical context in the rest\. Second, overfitting is*structural*: multi\-step prediction RMSE keeps improving long after the rewards/observations/controllable subspaces have drifted out of alignment, which is what hurts closed\-loop MPC\.

![Refer to caption](https://arxiv.org/html/2607.01736v1/figures/crof_analysis_top.png)Figure 5:CROF composite score over training\. Left: smoothed MPC return \(blue\) vs\. smoothed CROF\-A \(red, lower\-is\-better axis inverted\)\. Right: same for CROF\-B\. Vertical lines mark the smoothed MPC oracle \(blue dashed, epoch310310\), the MA\-7\-smoothed CROF minimum \(red dashed: CROF\-A at315315, CROF\-B at270270\), and the raw CROF minimum \(red dotted:280280for both variants\); see Table[2](https://arxiv.org/html/2607.01736#S4.T2)\.
### 4\.10Model\-Based Actor\-Critic Policy Training Performance

To test whether the CROF\-driven checkpoint selection \(Section[4\.9](https://arxiv.org/html/2607.01736#S4.SS9)\) translates into competitive downstream model\-based A2C policies, we train one A2C agent per world\-model \(WM\) checkpoint over a*purposive*sample of nine WMs that includes the CROF\-A/B and pure\-ROF picks alongside oracle, plateau, pre\-plateau, and three early\-training references \(per\-WM selection rules in Table[3](https://arxiv.org/html/2607.01736#S4.T3); WM6060is the raw\-MPC outlier from Section[4\.4](https://arxiv.org/html/2607.01736#S4.SS4)\)\. A uniform sweep over all100100WMs would be prohibitive\. All nine runs share the hyperparameters of Section[3\.3](https://arxiv.org/html/2607.01736#S3.SS3)\(10001000epochs, a checkpoint every5050epochs, seed1234512345\), and every saved actor is evaluated on100100deterministic episodes in the real environment \(seed sequence12345\+k12345\{\+\}k,k=0,…,99k=0,\\dots,99, max600600steps\) reporting mean, worst\-case, catastrophic failures \(return<−100\{<\}\{\-\}100\), and perfect landings \(return≥\+200\{\\geq\}\{\+\}200, the LunarLander “solved” threshold\)\.

Table[3](https://arxiv.org/html/2607.01736#S4.T3)reports, per WM checkpoint, the A2C checkpoint with the highest mean return \(“best\-by\-mean”, BBM\) and the checkpoint with the highest worst\-case return \(“safest\-by\-worst”, SBW\); SBW is the more practically useful pick for deployment, where worst\-case behavior matters\.

Best\-by\-mean A2C checkpointSafest\-by\-worst A2C checkpointWMSelection ruleEp\.MeanWorstCatasPerfectEp\.MeanWorstCatasPerfect\#safe5050early training550550\+178\.3\+178\.3−120\.8\-120\.8556464900900\+141\.3\+141\.3−74\.7\-74\.7047471/201/206060raw\-MPC outlier500500\+206\.3\+206\.3−157\.3\-157\.3557979600600\+182\.1\+182\.1−154\.7\-154\.78873730/200/20100100early training800800\+223\.3\\mathbf\{\+223\.3\}−108\.0\-108\.0118585200200\+65\.0\+65\.0−80\.2\-80\.2025253/203/20200200pre\-plateau reference10001000\+191\.4\+191\.4−184\.2\-184\.2337070300300\+149\.1\+149\.1−111\.3\-111\.31143430/200/20250250minjac\_rof\_combined\(raw\)300300\+141\.0\+141\.0−161\.2\-161\.2112626200200\+127\.3\+127\.3−104\.8\-104\.81128280/200/20𝟐𝟖𝟎\\mathbf\{280\}min CROF\-A, CROF\-B \(raw\)800800\+217\.5\\mathbf\{\+217\.5\}−178\.9\-178\.922𝟕𝟗\\mathbf\{79\}300300\+191\.3\\mathbf\{\+191\.3\}−39\.6\\mathbf\{\-39\.6\}0𝟕𝟏\\mathbf\{71\}2/202/20300300plateau midpoint850850\+205\.9\+205\.9−95\.5\\mathbf\{\-95\.5\}𝟎\\mathbf\{0\}7474600600\+138\.3\+138\.3−63\.1\-63\.104949𝟕/𝟐𝟎\\mathbf\{7/20\}310310smoothed\-MPC oracle10001000\+209\.6\+209\.6−131\.0\-131\.0227575400400\+196\.0\+196\.0−40\.4\-40\.4052523/203/20𝟑𝟏𝟓\\mathbf\{315\}min CROF\-A \(sm\. MA\-7\)800800\+178\.0\+178\.0−203\.5\-203\.5116363700700\+133\.1\+133\.1−72\.1\-72\.1047471/201/20

Table 3:A2C training outcomes across purposively sampled world\-model checkpoints,100100deterministic episodes per A2C checkpoint \(bold rows mark CROF metric\-driven picks\)\.Ep\.is the AC training epoch of the reported checkpoint;mean/worstare mean and worst\-case return;catascounts episodes with return<−100<\{\-\}100,perfectcounts≥\+200\\geq\{\+\}200;\#safeis the number of A2C checkpoints \(out of2020\) with0catastrophic episodes\. The*best\-by\-mean*\(BBM\) and*safest\-by\-worst*\(SBW\) blocks report, respectively, the A2C checkpoint with the highest mean and the checkpoint with the highest worst\-case return\.##### Analysis\.

A deployable WM is one whose BBM and SBW checkpoints both report a high mean\. By that joint criterion CROF’s raw pick WM280280wins: BBM\+217\.5\+217\.5and SBW\+191\.3\+191\.3with a∼26\{\\sim\}26\-point peak\-to\-safe gap, found offline with no environment evaluation\. The smoothed\-MPC oracle \(WM310310\) is comparable but requires the full MPC sweep to identify\. WM100100is the trap that the BBM column alone hides: highest BBM \(\+223\.3\+223\.3\), but its SBW checkpoint collapses to\+65\.0\+65\.0— a∼158\{\\sim\}158\-point peak\-to\-safe gap\. WM250250\(raw min ofjac\_rof\_combinedalone\) is the weakest deployable WM, showing that CROF’s auxiliary terms are doing real work: they nudge the selection out of the local\-ROF dip at WM250250into the structurally good region around WM280280\.

### 4\.11Model\-Free Actor\-Critic Policy

As a baseline, we train a model\-free A2C agent \(MF\-AC\) that interacts directly with the real environment\. Actor and critic are 3\-layer MLPs of width256256over the raw observation; training runs for10001000epochs at5050episodes per epoch \(50,00050\{,\}000total episodes,≈18\.4​M\\approx\{18\.4\\text\{M\}\}transitions, mean episode length≈368\\approx 368\), with learning rate3×10−43\{\\times\}10^\{\-4\},γ=0\.99\\gamma\{=\}0\.99,λ=0\.95\\lambda\{=\}0\.95, entropy decayed from0\.20\.2to0\.010\.01\.

MF\-AC is evaluated with the same100100\-episode protocol used for WM\-AC, but with LunarLander’s standard10001000\-step horizon \(WM\-AC and MPC use a600600\-step cap matching MPC’s per\-step inference budget\) to give the baseline its strongest evaluation\. MF\-AC peaks at AC epoch760760with mean return\+193\.04\\mathbf\{\+193\.04\}\(54/10054/100perfect,𝟎/𝟏𝟎𝟎\\mathbf\{0/100\}catastrophic, worst−35\.55\-35\.55\) and forms a reliable plateau over epochs∼610\{\\sim\}610–910910in the\+170\+170to\+195\+195band\. The head\-to\-head against WM\-AC is reported in Table[4](https://arxiv.org/html/2607.01736#S4.T4): MF\-AC has the tightest worst\-case, while WM\-AC has higher mean and more perfect landings at a fraction of the data cost\.

### 4\.12Data Efficiency Comparison

We measure training data efficiency cost for CEM\-MPC, WM\-AC, and MF\-AC\. We use real environment steps so they are compared on the same axis\. All model\-based methods share the same offline dataset \(180,916180\{,\}916steps over872872episodes\); MF\-AC’s cost is the cumulative transition count up to its best epoch760760\. Table[4](https://arxiv.org/html/2607.01736#S4.T4)summarizes the result\.

Table 4:Real environment\-step cost vs\. deterministic\-policy mean return \(100100episodes per checkpoint\)\. MPC and WM\-AC use a600600\-step cap matching MPC’s inference budget; MF\-AC uses the standard10001000\-stepLunarLander\-v3horizon under which it was trained\. All model\-based methods reuse the same872872\-episode offline dataset \(180,916180\{,\}916env\. steps\)\.Perfect==return≥\+200\{\\geq\}\{\+\}200\(“solved”\);Catast\.==return<−100\{<\}\{\-\}100\. Each WM\-AC row reports the BBM A2C checkpoint inside the corresponding CROF\-selected WM \(Table[3](https://arxiv.org/html/2607.01736#S4.T3)\)\. The CROF raw pick \(WM280280\) is the strongest model\-based result, beating MF\-AC’s peak by∼24\.5\{\\sim\}24\.5points at∼𝟔𝟓×\\bm\{\{\\sim\}65\\times\}fewer real environment interactions; MF\-AC is the only method with0/1000/100catastrophic but at two orders of magnitude higher data cost\.##### Analysis\.

The CROF raw WM\-AC pick \(WM280280, identified offline\) reaches peak mean\+217\.48\\mathbf\{\+217\.48\}and requires∼65\.3×\{\\sim\}\\mathbf\{65\.3\\times\}fewer real\-environment interactions than MF\-AC; CEM\-MPC on the same WM family already reaches\+166\.6\\mathbf\{\+166\.6\}atn=100n\{=\}100with no policy training\. At inference time, the trained A2C policy is∼1000×\{\\sim\}1000\\timescheaper to execute than CEM\-MPC, which runs44iterations of384384rollouts×25\\times\\,25steps at every real timestep\. This confirms substantially higher training\-data efficiency for the world\-model\-based methods\.

## 5Conclusion

We presented a study of validation\-time metrics for predicting the closed\-loop performance of a learned latent world model, using LunarLander\-v3 as the testbed and the*Composite Reward Observability Fraction*\(CROF\) as a candidate single\-number score for offline checkpoint selection\. On this task CROF tracks both CEM\-MPC and model\-based A2C performance, while standard criteria \(validation loss, multi\-step RMSE, empirical sensitivities\) instead pick deeply overfit checkpoints where MPC has poor closed\-loop performance\.

The CROF raw pick \(WM280280\) yields a model\-based A2C with peak mean return\+217\.5\\mathbf\{\+217\.5\}\(79/10079/100perfect landings\) —∼24\.5\{\\sim\}\\mathbf\{24\.5\}points above the model\-free A2C baseline \(\+193\.0\+193\.0\) at∼𝟔𝟓×\{\\sim\}\\mathbf\{65\\times\}fewer real\-environment interactions\. The smoothed\-MPC oracle \(WM310310, identifiable only via the full MPC sweep\) yields a comparable WM\-AC peak of\+209\.6\+209\.6, so on this task CROF matches an oracle pick at zero real\-environment\-selection cost\.

##### Scope and limitations\.

All quantitative claims here are on a single environment \(LunarLander\-v3\)\. The mechanistic argument for ROF leans on the specific structure of LunarLander’s reward \(per\-step shaping plus terminal events gated by simulator\-internal flags\), so we expect ROF to carry less signal on tasks whose reward is fully Markovian in\(ot,at\)\(o\_\{t\},a\_\{t\}\)— a Reacher control we report in Section[4\.8](https://arxiv.org/html/2607.01736#S4.SS8)confirms this directionally \(MLPR2=0\.9998R^\{2\}=0\.9998, ROF Spearmanρs=\+0\.143\\rho\_\{s\}=\{\+\}0\.143\)\. We therefore present CROF as a metric that targets a specific, but practically common, structural condition where rewards are non\-Markovian or shaped rather than as a domain\-agnostic metric\.

##### Future work\.

Natural extensions include cross\-environment validation \(continuous actions, higher\-dimensional observations, varied reward structures\), using a ROF\-like term as an auxiliary training regularizer, and developing an analogous offline metric for A2C\-policy checkpoint selection — an obvious gap in the current pipeline, where the final actor checkpoint still requires real\-environment evaluation\.

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## Appendix ASupplementary Figures

![Refer to caption](https://arxiv.org/html/2607.01736v1/figures/correlation_barchart.png)Figure 6:Per\-metric correlations with smoothed \(MA\-7\) MPC mean return for all4040metrics in the suite of Section[3\.4](https://arxiv.org/html/2607.01736#S3.SS4)\. Bars show Spearmanρs\\rho\_\{s\}; colours mark the strong \(\|ρs\|≥0\.30\{\|\\rho\_\{s\}\|\\geq 0\.30\}\), moderate \(0\.10≤\|ρs\|<0\.300\.10\\leq\{\|\\rho\_\{s\}\|<0\.30\}\), and weak \(\|ρs\|<0\.10\{\|\\rho\_\{s\}\|<0\.10\}\) bands used throughout the paper\. The ROF/CROF family dominates the strong\-correlation block, while all standard training and one\-step\-prediction metrics \(val\_loss,val\_recon,post\_\*\_rmse, etc\.\) sit in the weak band\. Numerical values for the most informative subset are listed in Table[1](https://arxiv.org/html/2607.01736#S4.T1)\.![Refer to caption](https://arxiv.org/html/2607.01736v1/figures/metrics_vs_mpc_comprehensive.png)Figure 7:Metrics vs\. MPC dashboard\. Standard metrics \(validation losses, open\-loop RMSEs, empirical sensitivities, Jacobian spectral radius and ranks\) are monotonic or flat through the late\-training collapse; only the ROF family \(third\-row right panel\) traces MPC’s rise–peak–collapse U\-shape\. Bottom row: scatter of the three core ROF sub\-metrics against MA\-7 MPC return with quadratic fits\.