Multi-Horizon Consistency as Geometry: When Latent Dynamics Contract, and When They Do Not

# When Latent Dynamics Contract, and When They Do Not
Source: [https://arxiv.org/html/2607.21645](https://arxiv.org/html/2607.21645)
## Multi\-Horizon Consistency as Geometry: When Latent Dynamics Contract, and When They Do Not

Kavya Bhand Department of Computer Engineering Vishwakarma Institute of Technology, Pune, India kavya\.bhand0806@gmail\.comAadi Joshi Department of Computer Engineering Vishwakarma Institute of Technology, Pune, India toaadijoshi@gmail\.com

###### Abstract

Multi\-horizon latent consistency is a common training knob in video predictors and world models, but practitioners rarely know what it does to transition geometry\. We treatλ\\lambda, the weight on multi\-step latent agreement, as a diagnostic control and measure an empirical expansion proxyL20,q95L\_\{20,\\mathrm\{q95\}\}together with horizon\-2020prediction errorE20E\_\{20\}\. On Moving\-MNIST \(n=6n\{=\}6seeds at the critical pair\), raisingλ\\lambdafrom0to0\.80\.8cutsL20L\_\{20\}from4\.96±2\.014\.96\{\\pm\}2\.01to1\.01±0\.061\.01\{\\pm\}0\.06\(pairedttp=0\.005p\{=\}0\.005, Wilcoxonp=0\.031p\{=\}0\.031\) and halvesE20E\_\{20\}\(0\.365→0\.1770\.365\{\\rightarrow\}0\.177, pairedttp=1\.1×10−13p\{=\}1\.1\{\\times\}10^\{\-13\}\)\. Four of six seeds crossL<1L\{<\}1atλ=0\.8\\lambda\{=\}0\.8\. The same loss does*not*produce populationL<1L\{<\}1on action\-conditioned Pendulum\-v1 or CartPole\-v1, nor on KTH Actions video, even whenE20E\_\{20\}improves\. An associational mediation analysis on MMNIST givesr^=0\.94\\hat\{r\}\{=\}0\.94\(95% CI\[0\.88,1\.00\]\[0\.88,1\.00\],n=27n\{=\}27,B=2000B\{=\}2000\);λ\\lambdawas not randomized\. Defensive checks \(architectural baselines, exogenous stress, WorldTest, MPC, scaling\) mostly support a narrow claim: soft consistency can push passive video toward a near\-contractive band, and that band is domain\-limited\. A stochastic\-forcing lawL^20≈1\.23\+1\.82​η\\hat\{L\}\_\{20\}\{\\approx\}1\.23\{\+\}1\.82\\etaatλ=0\.8\\lambda\{=\}0\.8\(bootstrap slope CI\[1\.73,1\.92\]\[1\.73,1\.92\],R2=0\.96R^\{2\}\{=\}0\.96\) unifies control domains on the same curve via calibratedηeff\\eta\_\{\\mathrm\{eff\}\}\. Complete joint slices atλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}\(30/3030/30cells,5​η×35\\eta\{\\times\}3seeds\) show comparable linearL20​\(η\)L\_\{20\}\(\\eta\)slopes \(≈1\.69\\approx 1\.69and≈2\.00\\approx 2\.00\); we do not fit a continuous\(λ,η\)\(\\lambda,\\eta\)surface\. We do not report DreamerV3 or TD\-MPC2 returns\.

Keywords:world models, latent consistency, Lipschitz expansion proxy, stochastic forcing, Moving\-MNIST, model\-based RL diagnostics

## 1Introduction

World models and joint\-embedding predictors often roll out farther than their training horizon\[[11](https://arxiv.org/html/2607.21645#bib.bib11),[12](https://arxiv.org/html/2607.21645#bib.bib12),[14](https://arxiv.org/html/2607.21645#bib.bib14),[18](https://arxiv.org/html/2607.21645#bib.bib18)\]\. When they fail, errors compound\. A practical question sits underneath those systems: if you turn up multi\-step consistency, what actually changes in the learned transition?

This paper does not propose a new architecture or a leaderboard entry\. It studies one training term,

ℒcons=∑k∈𝒦‖z^t\+k−zt\+ktf‖22,𝒦=\{1,3,5,10,15,20\},\\mathcal\{L\}\_\{\\mathrm\{cons\}\}=\\sum\_\{k\\in\\mathcal\{K\}\}\\bigl\\\|\\hat\{z\}\_\{t\+k\}\-z^\{\\mathrm\{tf\}\}\_\{t\+k\}\\bigr\\\|\_\{2\}^\{2\},\\quad\\mathcal\{K\}=\\\{1,3,5,10,15,20\\\},weighted byλ\\lambdaagainst reconstruction and one\-step losses, and asks whetherλ\\lambdamoves a finite\-sample expansion proxyL20,q95L\_\{20,\\mathrm\{q95\}\}across the reference lineL=1L\{=\}1\.

Practitioners already turn this knob\. What is missing is a domain\-aware account of what it does to transition geometry, when population contraction appears, and how that relates to action\-conditioned settings where planning actually happens\. We treat the study as a diagnostic: report where the proxy crossesL<1L\{<\}1, where it does not, and whether a single forcing parameter can place those regimes on one curve\.

The empirical picture is mixed in a useful way\. Passive Moving\-MNIST shows a sharp, seed\-replicated drop in bothL20L\_\{20\}andE20E\_\{20\}asλ\\lambdarises, with several seeds crossingL<1L\{<\}1nearλ=0\.8\\lambda\{=\}0\.8\. The same schedule on Pendulum, CartPole, and KTH tightens spectra or error without populationL<1L\{<\}1\. That split is the main result, and it motivates the stochastic\-forcing experiment: inject controllable noiseη\\etainto passive transitions and ask whether control domains sit on the sameL20​\(η\)L\_\{20\}\(\\eta\)curve\. They do, approximately linearly, which reframes the passive/active anecdotes as placements on one law rather than isolated failures\. Joint slices at two otherλ\\lambdavalues track the same qualitative rise inη\\eta, so the forcing story is not an artifact of fixingλ=0\.8\\lambda\{=\}0\.8\.

#### Contributions\.

1. 1\.A quantified MMNIST threshold: atλ=0\.8\\lambda\{=\}0\.8, meanL20L\_\{20\}sits near11\(4/64/6seeds below11\), with significant paired drops in bothL20L\_\{20\}andE20E\_\{20\}\(n=6n\{=\}6\)\.
2. 2\.A negative transfer result for action\-conditioned Gymnasium domains and for KTH:L<1L\{<\}1does not appear at the population level\.
3. 3\.Associational mediation \(r^=0\.94\\hat\{r\}\{=\}0\.94\) with an explicit non\-randomization caveat, plus run\-level sensitivity \(r^≈0\.80\\hat\{r\}\{\\approx\}0\.80\)\.
4. 4\.A stochastic\-forcing law at fixedλ=0\.8\\lambda\{=\}0\.8:L^20≈a\+b​η\\hat\{L\}\_\{20\}\\approx a\+b\\etawith bootstrap CIs onbb, placing Pendulum/CartPole via calibratedηeff\\eta\_\{\\mathrm\{eff\}\}\.
5. 5\.A joint\(λ,η\)\(\\lambda,\\eta\)boundary atλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}\(30/3030/30cells\): both slices rise approximately linearly inη\\etawith slopes near the primary law \(descriptive; no formalλ×η\\lambda\{\\times\}\\etainteraction test\)\.
6. 6\.Defensive experiments with locked multi\-seed numbers: WorldTest \(n=5n\{=\}5\), MPC \(n=3n\{=\}3, realλ=0\\lambda\{=\}0\), and scalingd∈\{16,32,64,128\}d\{\\in\}\\\{16,32,64,128\\\}\.

## 2Related work

We organize prior work by theme and state, for each, how the present diagnostic study relates\. The goal is not a leaderboard comparison but a map of*where*a training\-time consistency knob touches geometry, planning, and reliability\.

### 2\.1RSSM, Dreamer, and latent model\-based RL

Recurrent state\-space models \(RSSMs\) learn stochastic latents with reconstruction and rollout losses, then feed imagined trajectories to actors and critics\[[12](https://arxiv.org/html/2607.21645#bib.bib12),[13](https://arxiv.org/html/2607.21645#bib.bib13),[14](https://arxiv.org/html/2607.21645#bib.bib14),[11](https://arxiv.org/html/2607.21645#bib.bib11)\]\. Dreamer\-style agents treat the world model as a simulator for policy optimization; TD\-MPC2 later couples latent dynamics to gradient\-based MPC at scale\[[15](https://arxiv.org/html/2607.21645#bib.bib15)\]\. Contrastive and DeepMDP objectives also learn latent transitions for control\[[17](https://arxiv.org/html/2607.21645#bib.bib17),[10](https://arxiv.org/html/2607.21645#bib.bib10)\]\. These systems*implicitly*care about long\-horizon behavior \(policies are trained on multi\-step rollouts\) but they rarely report an explicit expansion statistic such asL20,q95L\_\{20,\\mathrm\{q95\}\}\. Our contribution is orthogonal: we hold architecture fixed \(GRU residual\) and ask what a*standalone*multi\-horizon consistency weightλ\\lambdadoes to a measurable local expansion proxy, without claiming a new agent or benchmark score\.

### 2\.2JEPA and representation\-space prediction

Joint\-embedding predictive architectures predict in representation space rather than pixel space\[[18](https://arxiv.org/html/2607.21645#bib.bib18),[3](https://arxiv.org/html/2607.21645#bib.bib3),[6](https://arxiv.org/html/2607.21645#bib.bib6)\]\. V\-JEPA 2 extends the family to action\-conditioned video\[[4](https://arxiv.org/html/2607.21645#bib.bib4)\]\. JEPAs stabilize training with architectural and objective choices \(masking, EMA targets, multi\-crop views\) that differ from our explicitℒcons\\mathcal\{L\}\_\{\\mathrm\{cons\}\}term\.Pendharkar \[[24](https://arxiv.org/html/2607.21645#bib.bib24)\]argue that pure predictive objectives can discard control\-relevant stochastic structure, a caution relevant when reading MMNIST contraction as universally desirable\. We include a partial exogenous\-digit stress test \(reportingL20L\_\{20\}/E20E\_\{20\}only\) to probe whether consistency collapses under injected stochasticity; it does not replicate their retention probes\. Our passive\-video results sit in the JEPA\-adjacent regime \(representation prediction without action labels\) but use a simpler GRU transition and a geometric diagnostic rather than a self\-supervised leaderboard\.

### 2\.3Lipschitz bounds, contraction, and stability

Hard stability constraints \(spectral normalization, antisymmetric transitions, stable RNN analyses\) bound gain directly\[[21](https://arxiv.org/html/2607.21645#bib.bib21),[8](https://arxiv.org/html/2607.21645#bib.bib8),[20](https://arxiv.org/html/2607.21645#bib.bib20),[22](https://arxiv.org/html/2607.21645#bib.bib22)\]\.Asadi et al\. \[[2](https://arxiv.org/html/2607.21645#bib.bib2)\]study model\-based RL through Wasserstein/Lipschitz model classes: they prove multi\-step and value\-error bounds that scale with the transition Lipschitz constant and show empirically that an*intermediate*hard clipkkon CartPole/Pendulum models maximizes returns when policies are co\-trained inside the learned model \(see §[9\.2](https://arxiv.org/html/2607.21645#S9.SS2)\)\. Our work uses a*soft*multi\-horizon agreement loss rather than a per\-layer norm cap, and we measure a finite\-sample expansion proxy rather than certifying‖f‖Lip\\\|f\\\|\_\{\\mathrm\{Lip\}\}\. The textbook horizon\-error recurrence \(Theorem[1](https://arxiv.org/html/2607.21645#Thmtheorem1)\) motivates monitoring whether an*estimated*proxy crossesL=1L\{=\}1; Proposition[1](https://arxiv.org/html/2607.21645#Thmproposition1)states whenL20,q95L\_\{20,\\mathrm\{q95\}\}is a valid finite\-sample statistic for localkk\-step gain \(Appendix[B](https://arxiv.org/html/2607.21645#A2)\)\. Concurrent theory lines pursue contractivity by other means: multi\-token prediction gradients\[[19](https://arxiv.org/html/2607.21645#bib.bib19)\], contractive global memory in video stacks\[[16](https://arxiv.org/html/2607.21645#bib.bib16)\], Lieb–Robinson locality bounds on pixel dynamics\[[1](https://arxiv.org/html/2607.21645#bib.bib1)\], and variable\-length latent supervision for planning\[[30](https://arxiv.org/html/2607.21645#bib.bib30)\]\. Those papers derive or enforce contraction structurally; we empirically*measure*whether a common training knob moves a proxy across a reference line, and document where it fails \(action\-conditioned control, KTH\)\.

### 2\.4Post\-hoc and test\-time diagnostics \(2026\)

A recent wave treats world\-model quality as a*diagnostic*rather than only a training loss\. ATM constructs an action\-consistency transfer matrix on frozen latent transitions to test whether action semantics survive rollouts\[[9](https://arxiv.org/html/2607.21645#bib.bib9)\]\.Ruan et al\. \[[25](https://arxiv.org/html/2607.21645#bib.bib25)\]score action–state compatibility in World Action Models on robotics simulators and use it for test\-time model selection\. AutumnBench WorldTest scores structural understanding via masked\-frame prediction and change detection\[[29](https://arxiv.org/html/2607.21645#bib.bib29)\]; we reuse that scoring recipe on our checkpoints but do not claim the full 43\-environment suite\. These tools answer “is this*trained*model reliable for planning or selection?” at evaluation time\. We instead varyλ\\lambda*during*training and trackL20,q95L\_\{20,\\mathrm\{q95\}\}andE20E\_\{20\}, producing a*training\-time*geometry curve\. The axes are complementary: ATM/Ruan\-style checks can screen a checkpoint; aλ\\lambdasweep explains how the checkpoint got there\. Neither substitutes for the other\.

### 2\.5Positioning

This paper is a mechanistic diagnostic study, not a new world model\. It contributes \(i\) a quantified passive\-video threshold inL20,q95L\_\{20,\\mathrm\{q95\}\}, \(ii\) negative transfer on action\-conditioned and natural\-video domains, \(iii\) associational mediation with explicit non\-randomization caveats, \(iv\) a stochastic\-forcing law linking passive and active regimes at fixedλ\\lambda, \(v\) complete joint\(λ,η\)\(\\lambda,\\eta\)slices atλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}, and \(vi\) defensive checks \(architectural baselines, exogenous stress, WorldTest, MPC, scaling\)\. It does not compete with Dreamer/TD\-MPC2 returns, certify global Lipschitz constants, or replace post\-hoc diagnostics\. The open question, why passive video crossesL<1L\{<\}1while active control does not, is developed in §[9\.1](https://arxiv.org/html/2607.21645#S9.SS1)and tested viaη\\etain §[6\.5](https://arxiv.org/html/2607.21645#S6.SS5)\.

#### What this paper is not\.

It is not a Dreamer or JEPA replacement; the architecture is fixed on purpose\. It is not a certified contraction theorem for video world models\. It does not claim thatL<1L\{<\}1improves planning returns—on Pendulum, our MPC result argues the opposite\. It does not identify a causalλ→L→E\\lambda\\to L\\to Eeffect \(mediation is associational\)\. If you want a new SOTA agent or a universal Lipschitz certificate, this is the wrong paper\. If you want to audit the geometry of a trained latent transition under a consistency weight, that is the target reader\.

## 3Preliminaries

#### Latent transition\.

Letzt∈𝒵⊆ℝdz\_\{t\}\\in\\mathcal\{Z\}\\subseteq\\mathbb\{R\}^\{d\}denote an encoded latent at timett\. A one\-step transition mapf:𝒵→𝒵f:\\mathcal\{Z\}\\to\\mathcal\{Z\}\(optionally conditioned on actionata\_\{t\}\) induces thekk\-step mapf\(k\)f^\{\(k\)\}\. Teacher\-forced latentszt\+ktfz^\{\\mathrm\{tf\}\}\_\{t\+k\}come from encoding observed frames or states; free\-run predictionsz^t\+k\\hat\{z\}\_\{t\+k\}roll outfffromztz\_\{t\}\.

#### Regimes\.

*Passive video*trainsffwithout actions \(Moving\-MNIST, KTH\)\.*Action\-conditioned control*trainsf​\(zt,at\)f\(z\_\{t\},a\_\{t\}\)on offline Gymnasium trajectories \(Pendulum\-v1, CartPole\-v1\)\. The same consistency weightλ\\lambdais applied in both regimes; the estimand below is regime\-agnostic, but the induced geometry need not be\.

#### Estimand\.

For pairs\(zi,zj\)\(z\_\{i\},z\_\{j\}\)drawn from a validation latent pool,

Ri​j\(k\)=‖f\(k\)​\(zi\)−f\(k\)​\(zj\)‖‖zi−zj‖,Lk,q95=Q^0\.95​\(\{Ri​j\(k\)\}\)\.R^\{\(k\)\}\_\{ij\}=\\frac\{\\\|f^\{\(k\)\}\(z\_\{i\}\)\-f^\{\(k\)\}\(z\_\{j\}\)\\\|\}\{\\\|z\_\{i\}\-z\_\{j\}\\\|\},\\qquad L\_\{k,\\mathrm\{q95\}\}=\\widehat\{Q\}\_\{0\.95\}\\bigl\(\\\{R^\{\(k\)\}\_\{ij\}\\\}\\bigr\)\.We reportL20,q95L\_\{20,\\mathrm\{q95\}\}\(writtenL20L\_\{20\}in tables\) and horizon\-2020prediction MSEE20E\_\{20\}\.L20,q95<1L\_\{20,\\mathrm\{q95\}\}\{<\}1means that on95%95\\%of sampled pairs the2020\-step map is finite\-difference non\-expansive; it is not a certificate thatsup‖Jf‖<1\\sup\\\|J\_\{f\}\\\|\{<\}1\(Appendix[B](https://arxiv.org/html/2607.21645#A2)\)\.

#### Stochastic forcing\.

To interpolate passive and active geometry we injectz←f​\(z\)\+η​σ⋆​ε/dz\\leftarrow f\(z\)\+\\eta\\,\\sigma^\{\\star\}\\varepsilon/\\sqrt\{d\},ε∼𝒩​\(0,I\)\\varepsilon\\sim\\mathcal\{N\}\(0,I\), during training at fixedλ\\lambda, and place control domains by calibratedηeff\\eta\_\{\\mathrm\{eff\}\}\(Section[6\.5](https://arxiv.org/html/2607.21645#S6.SS5)\)\.

## 4Method

### 4\.1Models

Passive video\.Convolutional encoder, GRU residual transitionz^t\+1=zt\+α​tanh⁡\(Δ​zt\)\\hat\{z\}\_\{t\+1\}=z\_\{t\}\+\\alpha\\tanh\(\\Delta z\_\{t\}\)withα=0\.5\\alpha\{=\}0\.5,d=64d\{=\}64, hidden width160160\[[26](https://arxiv.org/html/2607.21645#bib.bib26)\]\.

Action\-conditioned control\.MLP encoder of\(st,at\)\(s\_\{t\},a\_\{t\}\); transition consumes\(zt,at\)\(z\_\{t\},a\_\{t\}\)\. Offline random \(and, for CartPole, heuristic\) Gymnasium trajectories\[[28](https://arxiv.org/html/2607.21645#bib.bib28)\]\.

### 4\.2Loss and metrics

Total lossℒ=ℒrecon\+ℒ1​s​t​e​p\+λ​ℒcons\\mathcal\{L\}=\\mathcal\{L\}\_\{\\mathrm\{recon\}\}\+\\mathcal\{L\}\_\{\\mathrm\{1step\}\}\+\\lambda\\,\\mathcal\{L\}\_\{\\mathrm\{cons\}\}\. Sweeps useλ∈\{0,0\.2,…,1\.2\}\\lambda\\in\\\{0,0\.2,\\ldots,1\.2\\\}unless noted\.

For held\-out latent pairs the model rolls outK=20K\{=\}20steps and forms ratiosri​j\(k\)=‖f\(k\)​\(zi\)−f\(k\)​\(zj\)‖/‖zi−zj‖r\_\{ij\}^\{\(k\)\}=\\\|f^\{\(k\)\}\(z\_\{i\}\)\-f^\{\(k\)\}\(z\_\{j\}\)\\\|/\\\|z\_\{i\}\-z\_\{j\}\\\|\.L20,q95L\_\{20,\\mathrm\{q95\}\}is the9595th percentile of those ratios\. This is a finite\-sample local expansion proxy, not a certified global Lipschitz constant\[[20](https://arxiv.org/html/2607.21645#bib.bib20),[22](https://arxiv.org/html/2607.21645#bib.bib22)\]\.E20E\_\{20\}is MSE on stress\-split frames at horizon2020\.

### 4\.3Training

Default: AdamW, lr2×10−32\{\\times\}10^\{\-3\},2020MMNIST epochs /2525Pendulum epochs, seeds\{42,43,44\}\\\{42,43,44\\\}, with two extra MMNIST seeds atλ∈\{0,0\.8\}\\lambda\\in\\\{0,0\.8\\\}for paired tests \(n=6n\{=\}6\)\. Hyperparameters are listed in Appendix[C](https://arxiv.org/html/2607.21645#A3)\.

### 4\.4Evaluation protocol

Unless noted otherwise, each\(λ,seed\)\(\\lambda,\\mathrm\{seed\}\)cell is an independent training run from the same data split\.L20,q95L\_\{20,\\mathrm\{q95\}\}usesnpairs=240n\_\{\\mathrm\{pairs\}\}\{=\}240random chords from a held\-out stress or validation pool with a small number of loader batches \(default44–2525depending on stage; see Appendix[C](https://arxiv.org/html/2607.21645#A3)\)\. Action\-conditionedL20L\_\{20\}uses matched random action sequences for both members of a pair unless an ablation specifies zero or matched logged actions\. We report mean±\\pmstd across seeds and, for the MMNIST critical pair, pairedttand Wilcoxon tests\. Bootstrap confidence intervals for mediation and for theη\\eta\-law slope useB=2000B\{=\}2000resamples \(run\-level or seed\-block, respectively\)\. No hyperparameter search overλ\\lambdais claimed beyond the reported grid;λ=0\.8\\lambda\{=\}0\.8is selected as the MMNIST operating point where meanL20L\_\{20\}first sits near11with multiple seeds below11\.

### 4\.5Experimental design summary

The study is organized as a sequence of locked stages rather than an open\-ended leaderboard chase:

1. 1\.*Primary geometry curve*on Moving\-MNIST \(λ\\lambdagrid; critical pairn=6n\{=\}6\)\.
2. 2\.*Transfer negatives*on Pendulum, CartPole, and KTH under the sameλ\\lambdagrid\.
3. 3\.*Mechanism*via stochastic forcingη\\etaat fixedλ=0\.8\\lambda\{=\}0\.8\(primaryn=50n\{=\}50\)\.
4. 4\.*Defenses*: architectural baselines, exogenous digit, WorldTest\-style scoring, latent MPC, latent\-dimension scaling\.
5. 5\.*Joint*\(λ,η\)\(\\lambda,\\eta\)boundary atλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}\(30/3030/30cells; Appendix[I](https://arxiv.org/html/2607.21645#A9)\)\.

Every claim in the abstract is backed by a locked CSV underresults/master/and checked by the package validator \(Appendix[S](https://arxiv.org/html/2607.21645#A19)\)\.

Algorithm 1Train and evaluate a consistency\-weighted latent world model1:Dataset

𝒟\\mathcal\{D\}, weight

λ\\lambda, horizons

𝒦\\mathcal\{K\}, noise level

η\\eta\(default

0\)

2:Initialize encoder, transition

ff, decoder

3:forepoch

=1,…,E=1,\\ldots,Edo

4:forminibatch

B⊂𝒟B\\subset\\mathcal\{D\}do

5:Encode teacher\-forced latents

\{zt\}\\\{z\_\{t\}\\\}; optionally add noise of scale

η​σ⋆\\eta\\sigma^\{\\star\}
6:Compute

ℒrecon\+ℒ1​s​t​e​p\+λ​∑k∈𝒦‖z^t\+k−zt\+ktf‖22\\mathcal\{L\}\_\{\\mathrm\{recon\}\}\+\\mathcal\{L\}\_\{\\mathrm\{1step\}\}\+\\lambda\\sum\_\{k\\in\\mathcal\{K\}\}\\\|\\hat\{z\}\_\{t\+k\}\-z^\{\\mathrm\{tf\}\}\_\{t\+k\}\\\|\_\{2\}^\{2\}
7:AdamW step with gradient clipping

8:endfor

9:endfor

10:Encode validation pool; sample

npairsn\_\{\\mathrm\{pairs\}\}chords; compute

L20,q95L\_\{20,\\mathrm\{q95\}\}and

E20E\_\{20\}
11:returncheckpoint and metrics

## 5Theoretical background

We use two theoretical pieces: a horizon\-error recurrence under Lipschitz dynamics \(standard, but proved in full in Appendix[A](https://arxiv.org/html/2607.21645#A1)\), and a validity argument for the finite\-sample statisticL20,q95L\_\{20,\\mathrm\{q95\}\}\(Appendix[B](https://arxiv.org/html/2607.21645#A2)\)\.

### 5\.1Recurrence bound

Letf:𝒵→𝒵f:\\mathcal\{Z\}\\to\\mathcal\{Z\}be a latent transition map andz^t\+1=f​\(z^t\)\+ξt\\hat\{z\}\_\{t\+1\}=f\(\\hat\{z\}\_\{t\}\)\+\\xi\_\{t\}a learned rollout with per\-step disturbance‖ξt‖≤ϵ\\\|\\xi\_\{t\}\\\|\\leq\\epsilon\. Writeet=‖z^t−zt‖e\_\{t\}=\\\|\\hat\{z\}\_\{t\}\-z\_\{t\}\\\|for rollout error against a reference trajectory\.

###### Theorem 1\(Horizon error recurrence\)\.

IfffisLL\-Lipschitz on the rollout manifold ande0=0e\_\{0\}=0, then for horizonh≥1h\\geq 1,

eh≤Lh​e0\+ϵ​1−Lh1−L\(L≠1\),eh≤h​ϵ\(L=1\)\.e\_\{h\}\\leq L^\{h\}e\_\{0\}\+\\epsilon\\,\\frac\{1\-L^\{h\}\}\{1\-L\}\\qquad\(L\\neq 1\),\\qquad e\_\{h\}\\leq h\\epsilon\\qquad\(L=1\)\.\(1\)In particular, ifL<1L<1the disturbance term saturates ash→∞h\\to\\infty; ifL\>1L\>1, error can grow exponentially\.

The bound is classical in spirit\[[2](https://arxiv.org/html/2607.21645#bib.bib2),[5](https://arxiv.org/html/2607.21645#bib.bib5),[27](https://arxiv.org/html/2607.21645#bib.bib27)\]; Appendix[A](https://arxiv.org/html/2607.21645#A1)gives a step\-by\-step proof by induction\. We use it only to motivate monitoring an*estimated*expansion statistic: crossing a proxy below11is suggestive of non\-explosivekk\-step gain on the measured set, not a certificate thatsup‖Jf‖<1\\sup\\\|J\_\{f\}\\\|<1\.

### 5\.2WhatL20,q95L\_\{20,\\mathrm\{q95\}\}estimates

For held\-out latentsziz\_\{i\}andkk\-step mapf\(k\)f^\{\(k\)\}, define pair ratiosri​j\(k\)=‖f\(k\)​\(zi\)−f\(k\)​\(zj\)‖/‖zi−zj‖r\_\{ij\}^\{\(k\)\}=\\\|f^\{\(k\)\}\(z\_\{i\}\)\-f^\{\(k\)\}\(z\_\{j\}\)\\\|/\\\|z\_\{i\}\-z\_\{j\}\\\|\. Our metricL20,q95L\_\{20,\\mathrm\{q95\}\}is the9595th percentile of\{ri​j\(20\)\}\\\{r\_\{ij\}^\{\(20\)\}\\\}overnpairsn\_\{\\mathrm\{pairs\}\}random pairs from validation latents \(implementation:src/metrics\.py\)\.

###### Proposition 1\(Proxy validity \(informal\)\)\.

SupposeffisLL\-Lipschitz on the support of the validation latent distribution\. Then every sampled ratio satisfiesri​j\(k\)≤Lkr\_\{ij\}^\{\(k\)\}\\leq L^\{k\}\. If additionallyffisC1C^\{1\}, pairs are drawn with‖zi−zj‖≤δ\\\|z\_\{i\}\-z\_\{j\}\\\|\\leq\\deltafor smallδ\\delta, and rollout actions are fixed across the pair \(passive video or matched\-action protocol\), thenri​j\(k\)r\_\{ij\}^\{\(k\)\}concentrates around the directional gain‖\(Jf​\(zi\)k​\(zj−zi\)\)‖/‖zj−zi‖\\\|\(J\_\{f\}\(z\_\{i\}\)^\{k\}\(z\_\{j\}\-z\_\{i\}\)\)\\\|/\\\|z\_\{j\}\-z\_\{i\}\\\|asδ→0\\delta\\to 0\. The empirical9595th percentile is a consistent estimator of the9595th quantile ofri​j\(k\)r\_\{ij\}^\{\(k\)\}under i\.i\.d\. pair resampling asnpairs→∞n\_\{\\mathrm\{pairs\}\}\\to\\infty\.

Appendix[B](https://arxiv.org/html/2607.21645#A2)states coverage conditions, relateskk\-step ratios to composed Jacobians, and connects the statistic to a one\-step spectral\-radius probe\. A short Jacobian power\-iteration check on MMNIST \(Appendix[L](https://arxiv.org/html/2607.21645#A12)\) shows bothL20,q95L\_\{20,\\mathrm\{q95\}\}and mean one\-step spectral radius drop whenλ\\lambdarises \(1\.29→1\.071\.29\\to 1\.07and1\.29→1\.021\.29\\to 1\.02at six epochs\), consistent with Proposition[1](https://arxiv.org/html/2607.21645#Thmproposition1)but not certifying a global Lipschitz constant\.

## 6Results

### 6\.1Moving\-MNIST: a rapidλ\\lambdathreshold

Table[1](https://arxiv.org/html/2607.21645#S6.T1)and Figure[1](https://arxiv.org/html/2607.21645#S6.F1)summarize2727MMNIST runs, withn=6n\{=\}6atλ∈\{0,0\.8\}\\lambda\\in\\\{0,0\.8\\\}\.L20L\_\{20\}falls from4\.96±2\.014\.96\{\\pm\}2\.01to1\.01±0\.061\.01\{\\pm\}0\.06\(pairedttp=0\.005p\{=\}0\.005; Wilcoxonp=0\.031p\{=\}0\.031\)\.E20E\_\{20\}falls from0\.365±0\.00070\.365\{\\pm\}0\.0007to0\.177±0\.00030\.177\{\\pm\}0\.0003\(pairedttp=1\.1×10−13p\{=\}1\.1\{\\times\}10^\{\-13\}; Wilcoxonp=0\.031p\{=\}0\.031\)\. Seeds43,45,46,4743,45,46,47sit belowL=1L\{=\}1atλ=0\.8\\lambda\{=\}0\.8\(4/64/6\)\. MeanL20L\_\{20\}at that setting is on the boundary rather than deep in a contractive regime\.

Call this an*empirical regime change*: a rapid threshold in measuredL20L\_\{20\}andE20E\_\{20\}asλ\\lambdaincreases\. No non\-analytic statistical\-mechanics phase transition is claimed\.

Table 1:Moving\-MNISTλ\\lambda\-sweep \(mean±\\pmstd\)\.n=6n\{=\}6atλ∈\{0,0\.8\}\\lambda\\in\\\{0,0\.8\\\}; otherwisen=3n\{=\}3\.![Refer to caption](https://arxiv.org/html/2607.21645v1/x1.png)

![Refer to caption](https://arxiv.org/html/2607.21645v1/x2.png)

Figure 1:Left: MMNISTL20L\_\{20\}andE20E\_\{20\}vs\.λ\\lambda\. Right: passive MMNIST vs\. action\-conditioned Pendulum \(mean±\\pmstd; dashed lineL=1L\{=\}1\)\.
### 6\.2Pendulum and CartPole: tightening withoutL<1L\{<\}1

On Pendulum\-v1,L20L\_\{20\}decreases from2\.51±0\.172\.51\{\\pm\}0\.17atλ=0\\lambda\{=\}0\(n=5n\{=\}5\) to1\.74±0\.131\.74\{\\pm\}0\.13atλ=0\.8\\lambda\{=\}0\.8\(n=3n\{=\}3\), then stays near1\.781\.78atλ=1\.2\\lambda\{=\}1\.2\. No seed crossesL<1L\{<\}1\.E20E\_\{20\}is already near10−410^\{\-4\}, so error is a weak phase detector\. Action\-protocol ablations atλ=0\.8\\lambda\{=\}0\.8\(random / zero / matched actions\) leaveL20L\_\{20\}in\{1\.73,1\.81,1\.85\}\\\{1\.73,1\.81,1\.85\\\}\(Appendix[E](https://arxiv.org/html/2607.21645#A5)\)\.

CartPole\-v1 shows the same qualitative split:L20L\_\{20\}stays expansive at everyλ\\lambdawithn=5n\{=\}5\(2\.96±0\.262\.96\{\\pm\}0\.26,3\.18±0\.243\.18\{\\pm\}0\.24,3\.17±0\.423\.17\{\\pm\}0\.42,2\.99±0\.202\.99\{\\pm\}0\.20atλ∈\{0,0\.4,0\.8,1\.2\}\\lambda\\in\\\{0,0\.4,0\.8,1\.2\\\}\) whileE20E\_\{20\}drops onceλ≥0\.4\\lambda\{\\geq\}0\.4\(Appendix[M](https://arxiv.org/html/2607.21645#A13)\)\. On dm\_control cartpole\-swingup \(offline random trajectories, same GRU residual backbone\), a CPU sweep yieldsλ=0\\lambda\{=\}0:L20=1\.41±0\.01L\_\{20\}\{=\}1\.41\{\\pm\}0\.01\(n=3n\{=\}3\);λ=0\.4\\lambda\{=\}0\.4:1\.50±0\.011\.50\{\\pm\}0\.01\(n=3n\{=\}3\);λ=0\.8\\lambda\{=\}0\.8:1\.55±0\.131\.55\{\\pm\}0\.13\(n=3n\{=\}3\);λ=1\.2\\lambda\{=\}1\.2:1\.55±0\.061\.55\{\\pm\}0\.06\(n=2n\{=\}2; seed4444not completed\)\. No populationL<1L\{<\}1appears on this grid, matching Gymnasium CartPole\-v1 qualitatively \(tighter absoluteL20L\_\{20\}, still expansive\)\. We treat this as a recognized\-control sanity check rather than a Dreamer\-scale benchmark\.

### 6\.3KTH Actions

On real KTH clips \(n=3n\{=\}3, seq\_len1616\),L20L\_\{20\}tightens from3\.32±2\.573\.32\{\\pm\}2\.57atλ=0\\lambda\{=\}0to1\.54±0\.661\.54\{\\pm\}0\.66atλ=1\.2\\lambda\{=\}1\.2\. Every seed decreases, seed4444remains an expansive outlier, and no seed crossesL<1L\{<\}1\.E20E\_\{20\}stays near0\.050\.05\. A noisy three\-digit MNIST proxy \(Appendix[K](https://arxiv.org/html/2607.21645#A11)\) is retained only as a stress test; KTH is the natural\-video result\.

### 6\.4Latent dimension

Ford∈\{16,32,128\}d\\in\\\{16,32,128\\\}we sweepλ∈\{0,0\.6,0\.8\}\\lambda\\in\\\{0,0\.6,0\.8\\\}withn=3n\{=\}3;d=64d\{=\}64is the main MMNIST table\. Atd=16d\{=\}16, meanL20=0\.91L\_\{20\}\{=\}0\.91already atλ=0\\lambda\{=\}0\(2/32/3seedsL<1L\{<\}1\)\. Atd=32d\{=\}32,L20L\_\{20\}falls3\.47→0\.893\.47\{\\rightarrow\}0\.89byλ=0\.6\\lambda\{=\}0\.6\(1\.011\.01atλ=0\.8\\lambda\{=\}0\.8\)\. Atd=128d\{=\}128,L20L\_\{20\}falls4\.96→1\.144\.96\{\\rightarrow\}1\.14and never crossesL<1L\{<\}1\. Larger latents need strongerλ\\lambdato approach the contractive band\. This finite\-size pattern is consistent with an under\-constrained residual map in highdd: soft multi\-horizon agreement must fight more directions before the9595th percentile of chord ratios drops below11\.

### 6\.5Stochastic forcing: a linear expansion law

To unify the passive/active split, we inject action\-like noise into passive MMNIST transitions during training:z←f​\(z\)\+η​σ⋆​ε/dz\\leftarrow f\(z\)\+\\eta\\,\\sigma^\{\\star\}\\varepsilon/\\sqrt\{d\}withε∼𝒩​\(0,I\)\\varepsilon\\sim\\mathcal\{N\}\(0,I\), fixedλ=0\.8\\lambda\{=\}0\.8, andσ⋆=0\.5\\sigma^\{\\star\}\{=\}0\.5\. Primary GPU sweep:η∈\{0,0\.1,…,1\.2\}\\eta\\in\\\{0,0\.1,\\ldots,1\.2\\\}\(10 points\)×\\times5 seeds \(n=50n\{=\}50runs, 16 epochs\)\. MeanL20,q95L\_\{20,\\mathrm\{q95\}\}increases approximately linearly,L^20≈1\.23\+1\.82​η\\hat\{L\}\_\{20\}\\approx 1\.23\+1\.82\\eta\(block\-bootstrap 95% CI on slope\[1\.73,1\.92\]\[1\.73,1\.92\],R2=0\.96R^\{2\}\{=\}0\.96; Figure[2](https://arxiv.org/html/2607.21645#S6.F2)\)\.

![Refer to caption](https://arxiv.org/html/2607.21645v1/x3.png)Figure 2:Stochastic forcing atλ=0\.8\\lambda\{=\}0\.8: meanL20,q95L\_\{20,\\mathrm\{q95\}\}vs\.η\\etaon MMNIST \(n=5n\{=\}5seeds per point; SEM bars\)\. Dotted line: least\-squares fit\. Vertical dotted lines: calibratedηeff\\eta\_\{\\mathrm\{eff\}\}for Pendulum, CartPole, and KTH\.We do*not*observe a mean\-levelL=1L\{=\}1upcrossing on this grid \(L^20​\(η=0\)=1\.10\\hat\{L\}\_\{20\}\(\\eta\{=\}0\)\{=\}1\.10, per\-η\\eta95% CI\[0\.95,1\.26\]\[0\.95,1\.26\]\), but calibrated effective noise levels place Pendulum \(ηeff≈0\.53\\eta\_\{\\mathrm\{eff\}\}\{\\approx\}0\.53\) and CartPole \(ηeff≈1\.0\\eta\_\{\\mathrm\{eff\}\}\{\\approx\}1\.0\) at predictedL^20≈2\.2\\hat\{L\}\_\{20\}\\approx 2\.2and3\.13\.1, consistent with populationL\>1L\{\>\}1in those domains\. KTH is the exception under the same calibration:ηeff≈0\\eta\_\{\\mathrm\{eff\}\}\{\\approx\}0by construction \(no actions\), yetL20L\_\{20\}stays expansive; we treat that as a natural\-video residual outside the forcing law rather than a contradiction of the Pendulum/CartPole placements\. Primary bootstrap CIs \(n=5n\{=\}5seeds perη\\eta\) are reported ineta\_law\_bootstrap\.json\. Joint boundary slices atλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}are complete \(30/3030/30cells; Appendix[I](https://arxiv.org/html/2607.21645#A9)\) and show comparable linearL20​\(η\)L\_\{20\}\(\\eta\)slopes; we do not claim a denser continuous\(λ,η\)\(\\lambda,\\eta\)surface or a finerη\\etagrid on the primary curve\.

## 7Associational mediation

We fit Baron–Kenny paths on z\-scoredλ\\lambda,log⁡L20\\log L\_\{20\}, andlog⁡E20\\log E\_\{20\}with DAGλ→L→E\\lambda\\to L\\to Eand a directλ→E\\lambda\\to Eedge\[[7](https://arxiv.org/html/2607.21645#bib.bib7),[23](https://arxiv.org/html/2607.21645#bib.bib23)\]\.

![Refer to caption](https://arxiv.org/html/2607.21645v1/x4.png)Figure 3:Associational mediation DAG used for the path\-ratio analysis\. Solid arrows:λ→L20→E20\\lambda\\to L\_\{20\}\\to E\_\{20\}; dashed: directλ→E20\\lambda\\to E\_\{20\}\.Point estimates use sevenλ\\lambda\-averaged means\. Confidence intervals use run\-resampled bootstrap \(B=2000B\{=\}2000\): resample2727runs, re\-aggregate, recompute paths\.

Primary result:r^=0\.94\\hat\{r\}\{=\}0\.94, 95% CI\[0\.88,1\.00\]\[0\.88,1\.00\]\. Run\-level sensitivity on all2727tuples:r^≈0\.80\\hat\{r\}\{\\approx\}0\.80, CI\[0\.65,0\.95\]\[0\.65,0\.95\]\. Becauseλ\\lambdais a training hyperparameter rather than a randomized intervention, these numbers are associational\. They do not identify a causal effect in the Pearl sense\.

Interpretation is deliberately cautious\. A large path ratio is consistent with the story thatλ\\lambdaassociates with geometry and that geometry associates with long\-horizon error on MMNIST\. It is not a license to say “raiseλ\\lambdato contractLL, thenEEfalls\.” Randomizingλ\\lambda\(orη\\eta\) inside a fixed architecture would be the next design step for causal language \(Section[12](https://arxiv.org/html/2607.21645#S12)\)\.

## 8Defensive experiments

### 8\.1What we do not compare

DreamerV3 DMC returns, TD\-MPC2 multi\-task returns, RSSM stochastic\-latent variants, and a full AutumnBench leaderboard are out of scope\[[14](https://arxiv.org/html/2607.21645#bib.bib14),[15](https://arxiv.org/html/2607.21645#bib.bib15),[29](https://arxiv.org/html/2607.21645#bib.bib29)\]\. Those omissions are intentional\. The paper is a diagnostic study ofλ\\lambda\. Comparing against agent returns would change the estimand from geometry to control performance and would require a different experimental budget\.

### 8\.2Architectural baselines

Atλ=0\.8\\lambda\{=\}0\.8on MMNIST \(n=3n\{=\}3\), soft consistency \(L20=1\.04±0\.08L\_\{20\}\{=\}1\.04\{\\pm\}0\.08\) matches spectral normalization \(1\.05±0\.161\.05\{\\pm\}0\.16\) and beats AntisymmetricRNN \(5\.22±0\.385\.22\{\\pm\}0\.38\), which over\-constrains the dynamics \(Figure[4](https://arxiv.org/html/2607.21645#S8.F4), left\)\. Soft consistency is therefore competitive with a hard spectral constraint on this passive\-video setting, without requiring a custom transition parameterization\. We do not claim superiority outside MMNIST; the point is thatλ\\lambdais not a strictly weaker knob than spectral normalization for theL20L\_\{20\}proxy here\.

### 8\.3Exogenous digit

Following the spirit ofPendharkar \[[24](https://arxiv.org/html/2607.21645#bib.bib24)\], a digit with resampled random velocity is added and onlyL20L\_\{20\}/E20E\_\{20\}are tracked\. Atλ=1\.2\\lambda\{=\}1\.2, exogenousL20=0\.97±0\.05L\_\{20\}\{=\}0\.97\{\\pm\}0\.05and controlL20=1\.17±0\.20L\_\{20\}\{=\}1\.17\{\\pm\}0\.20\(n=3n\{=\}3\)\. No MSE/LLcollapse appears\. That is not a refutation of exogenous discard in JEPAs; feature\-retention probes are absent\.

### 8\.4WorldTest\-style scoring

In\-domain MMNIST behavioral scores \(n=5n\{=\}5both arms\):0\.41±0\.080\.41\{\\pm\}0\.08atλ=0\\lambda\{=\}0to1\.06±0\.091\.06\{\\pm\}0\.09atλ=0\.8\\lambda\{=\}0\.8\(pairedΔ\+0\.64\\Delta\{\+\}0\.64\)\. Masked\-frame prediction stays near chance; change detection drives the gain\. On a grid\-world pixel adapter \(OOD\), scores move1\.88±0\.35→1\.17±0\.521\.88\{\\pm\}0\.35\\rightarrow 1\.17\{\\pm\}0\.52\(Δ−0\.70\\Delta\{\-\}0\.70\): consistency helps in\-domain CD and hurts the OOD adapter\. AutumnBench leaderboard evidence is not claimed\.

### 8\.5Latent MPC \(negative sweet\-spot\)

Random\-shooting MPC on Pendulum \(H=10H\{=\}10,K=32K\{=\}32,2020episodes, seeds\{42,43,44\}\\\{42,43,44\\\}\) with realλ=0\\lambda\{=\}0checkpoints: mean returns−713±62\-713\{\\pm\}62\(λ=0\\lambda\{=\}0\),−756±12\-756\{\\pm\}12\(λ=0\.8\\lambda\{=\}0\.8\),−768±23\-768\{\\pm\}23\(λ=1\.2\\lambda\{=\}1\.2\)\. Best mean atλ=0\\lambda\{=\}0;λ=0\.8\\lambda\{=\}0\.8wins on1/31/3seeds\. There is no planning sweet spot at the MMNISTλ\\lambdathreshold\. That ranking differs from an earlier single\-seed run that used aλ=0\.2\\lambda\{=\}0\.2proxy for the expansive regime; the multi\-seed result with trueλ=0\\lambda\{=\}0supersedes it\. See §[9\.2](https://arxiv.org/html/2607.21645#S9.SS2)for reconciliation withAsadi et al\. \[[2](https://arxiv.org/html/2607.21645#bib.bib2)\]\.

The negative MPC result is load\-bearing for the paper’s framing\. IfL<1L\{<\}1on passive video were a sufficient condition for better latent planning on control, we would expectλ=0\.8\\lambda\{=\}0\.8to help on Pendulum\. It does not under this planner\. That keeps the diagnostic honest: geometry and planning utility are separable claims\.

![Refer to caption](https://arxiv.org/html/2607.21645v1/x5.png)

![Refer to caption](https://arxiv.org/html/2607.21645v1/x6.png)

Figure 4:Left: architectural baselines atλ=0\.8\\lambda\{=\}0\.8\. Right: Pendulum latent\-MPC returns \(n=3n\{=\}3seeds,2020episodes each\)\.

## 9Discussion

The usable claim is narrow: soft consistency contracts passive\-video geometry under theL20L\_\{20\}proxy, and that effect does not automatically transfer to action\-conditioned or natural\-video settings\.

### 9\.1Passive versus active contraction

Moving\-MNIST is*passive*: transitions are learned from unlabeled video, actions are absent at train time, and the consistency term aligns teacher\-forced multi\-step latents on a low\-dimensional digit manifold\. Raisingλ\\lambdasharply movesL20,q95L\_\{20,\\mathrm\{q95\}\}toward11and cutsE20E\_\{20\}, four of six seeds crossL<1L\{<\}1atλ=0\.8\\lambda\{=\}0\.8\. Pendulum, CartPole, and KTH are*active*or*natural\-video*regimes: action channels \(or richer appearance statistics\) enter the transition, offline data are simulator rollouts or human motion clips, and the sameλ\\lambdatightens spectra or error without populationL<1L\{<\}1\.

We read this as a domain boundary, not a failed hyperparameter sweep\. Passive video offers a nearly deterministic latent flow on which multi\-horizon agreement can act like implicit spectral regularization, pushing localkk\-step gain toward a non\-expansive band \(Proposition[1](https://arxiv.org/html/2607.21645#Thmproposition1)\)\. Once actions \(or exogenous visual factors\) inject directions that must be preserved for control or realism, the same loss still improves one\-step or short\-horizon fit in places but no longer collapses the9595th percentile ofkk\-step ratios below11\.Pendharkar \[[24](https://arxiv.org/html/2607.21645#bib.bib24)\]warn that aggressive predictive objectives can discard stochastic degrees of freedom needed downstream; our action\-conditioned negatives are consistent with that picture, though we do not run their retention probes\.

Why might passive video cross and active control not? Three mechanisms are plausible and not mutually exclusive:

1. 1\.*Identifiability\.*Passive digits lie on a low\-curvature manifold where contraction and prediction align; control latents must encode torque\-relevant state, enlarging local Jacobian variation\.
2. 2\.*Loss competition\.*ℒcons\\mathcal\{L\}\_\{\\mathrm\{cons\}\}can dominate passive training at highλ\\lambda; on Pendulum/CartPole, near\-zeroE20E\_\{20\}already decouples error fromL20L\_\{20\}, so geometry and MSE need not move together\.
3. 3\.*Action routing\.*Matched\-action ablations leaveL20≈1\.7L\_\{20\}\\approx 1\.7–1\.91\.9atλ=0\.8\\lambda\{=\}0\.8\(Appendix[E](https://arxiv.org/html/2607.21645#A5)\), suggesting action\-conditioned Jacobians resist the same isotropic shrinkage that passive video admits\.

Section[6\.5](https://arxiv.org/html/2607.21645#S6.SS5)tests \(i\)–\(iii\) via controllableη\\eta: a single linear law with bootstrap confidence bands places passive video nearη=0\\eta\{=\}0and control domains at higherηeff\\eta\_\{\\mathrm\{eff\}\}, reducing four disconnected anecdotes to one reported scaling relation\.

### 9\.2Reconciling planning sweet spots withAsadi et al\. \[[2](https://arxiv.org/html/2607.21645#bib.bib2)\]

Asadi et al\. \[[2](https://arxiv.org/html/2607.21645#bib.bib2)\]bound multi\-step Wasserstein error and value error in terms of a model Lipschitz constant, then empirically cap weight norms by a hard boundkkwhile learning CartPole \(15×10315\{\\times\}10^\{3\}tuples\) and Pendulum \(10410^\{4\}tuples\) transition models\. For eachkk, they select the median cross\-validation model and train actor\-critic / deterministic policy gradient inside the learned model, evaluating returns on the true environment\. Figure 6 of their paper shows an intermediatekkmaximizing average return, a planning sweet spot on spectral gain during coupled model–policy learning\.

Our Pendulum latent\-MPC experiment is deliberately decoupled:λ\\lambdais varied only at train time, checkpoints are frozen, and a random\-shooting planner searches action sequences in latent space \(H=10H\{=\}10,K=32K\{=\}32,n=3n\{=\}3seeds\)\. Mean returns are best atλ=0\\lambda\{=\}0\(−713±62\-713\{\\pm\}62\) and worsen atλ=0\.8\\lambda\{=\}0\.8and1\.21\.2\(−756±12\-756\{\\pm\}12,−768±23\-768\{\\pm\}23\); there is no sweet spot at the MMNIST contraction threshold\.

These results are compatible rather than contradictory:

- •Different knobs: Asadi controlkk, a hard per\-layer Lipschitz cap on the transition network; we controlλ\\lambda, a soft penalty on multi\-horizon latent agreement\. Either can shrink local gain without tracking the other\.
- •Different planning loops: their sweet spot arises when policy gradients exploit a model whose gain was tuned jointly with control; our negative result shows that a passive\-video\-style contraction point need not help a fixed latent planner on a control checkpoint\.
- •Different readouts: they optimize environment return; we report an expansion proxyL20,q95L\_\{20,\\mathrm\{q95\}\}and a crude MPC score\. Improved geometry on MMNIST does not license transferringλ=0\.8\\lambda\{=\}0\.8to Pendulum planning\.

Takeaway for practitioners: spectral regularization can help planning when the regularizer, model, and planner are co\-designed; a training\-timeλ\\lambdathat crossesL<1L\{<\}1on passive video is neither necessary nor sufficient for latent MPC gains on the control domains tested here\.

### 9\.3Diagnostics, mediation, and open questions

ATM and future\-compatibility diagnostics\[[9](https://arxiv.org/html/2607.21645#bib.bib9),[25](https://arxiv.org/html/2607.21645#bib.bib25)\]screen*trained*models for action\-consistent rollouts; concurrent theory work enforces or analyzes contractivity via multi\-token losses, memory gating, or locality bounds\[[19](https://arxiv.org/html/2607.21645#bib.bib19),[16](https://arxiv.org/html/2607.21645#bib.bib16),[1](https://arxiv.org/html/2607.21645#bib.bib1),[30](https://arxiv.org/html/2607.21645#bib.bib30)\]\. Ourλ\\lambdasweep is a*training\-time*geometry curve on a fixed GRU backbone\. Use all three axes when auditing a world model: post\-hoc action/future tests, structural contractivity theory, and a consistency ablation withL20,q95L\_\{20,\\mathrm\{q95\}\}\.

Associational mediation \(r^=0\.94\\hat\{r\}\{=\}0\.94on MMNIST\) supports a coherent story:λ\\lambdaassociates with lowerL20L\_\{20\}and lowerE20E\_\{20\}, butλ\\lambdawas not randomized, so causal language is avoided\. The passive/active split is partly unified by the stochastic\-forcing law \(§[6\.5](https://arxiv.org/html/2607.21645#S6.SS5)\): soft consistency is a domain\-limited geometry diagnostic—useful on passive video at lowηeff\\eta\_\{\\mathrm\{eff\}\}, and misleading if exported wholesale to control without remeasurement on the same curve\. Joint slices atλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}\(Appendix[I](https://arxiv.org/html/2607.21645#A9)\) track the same qualitative rise inη\\eta, which argues against treating the primaryλ=0\.8\\lambda\{=\}0\.8slope as a one\-off\.

## 10Limitations

Sample sizes are modest:n=6n\{=\}6at the critical MMNIST pair,n=5n\{=\}5on CartPole \(allλ\\lambda\) and on Pendulum atλ=0\\lambda\{=\}0,n=4n\{=\}4on Pendulum atλ=0\.4\\lambda\{=\}0\.4,n=3n\{=\}3elsewhere unless noted, andn=5n\{=\}5seeds on the primaryη\\etagrid\. For the MMNIST pairedL20L\_\{20\}drop \(4\.96→1\.014\.96\\to 1\.01\), the observed effect is large relative to within\-seed variability \(pairedttp=0\.005p\{=\}0\.005\), so the critical\-pair comparison is not underpowered for detecting a large shift; smaller effects elsewhere would need more seeds\.L20,q95L\_\{20,\\mathrm\{q95\}\}is a local proxy whose validity conditions are in Appendix[B](https://arxiv.org/html/2607.21645#A2); it is notsup‖Jf‖\\sup\\\|J\_\{f\}\\\|\. Mediation is associational\. Benchmarks are Moving\-MNIST, Pendulum\-v1, CartPole\-v1, KTH, and dm\_control cartpole\-swingup \(CPU sweep:11/1211/12cells locked;λ=1\.2\\lambda\{=\}1\.2missing seed4444\)\. WorldTest uses an AutumnBench\-style scorer on our checkpoints, not the 43\-environment suite\. MPC is random shooting only\. The exogenous test lacks retention probes\. Architecture is GRU\-only\. The joint\(λ,η\)\(\\lambda,\\eta\)boundary is complete atλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}\(n=3n\{=\}3per cell; Appendix[I](https://arxiv.org/html/2607.21645#A9)\): both slices rise approximately linearly inη\\eta, with slopes comparable to the primaryλ=0\.8\\lambda\{=\}0\.8law; we still do not fit a full continuous surface overλ\\lambda\. Logged compute is on the order of5050GPU\-hours plus multi\-day CPU fill\-ins \(Appendix compute table\)\.

### 10\.1Threats to validity

#### Internal\.

λ\\lambdais a training hyperparameter, not a randomized intervention, soλ→L→E\\lambda\\to L\\to Epaths are associational\.L20,q95L\_\{20,\\mathrm\{q95\}\}depends on pair sampling and validation pools; Appendix[B](https://arxiv.org/html/2607.21645#A2)lists coverage conditions\.

#### External\.

Passive\-video contraction need not transfer to richer video \(KTH already fails at the population level\) or to Dreamer\-scale agents with larger latents and coupled planners\.

#### Construct\.

L20<1L\_\{20\}\{<\}1is a finite\-difference non\-expansiveness event, not a planning\-quality certificate; our MPC negative result makes that distinction operational\.

## 11Conclusion

Raising multi\-horizon consistency on Moving\-MNIST produces a clear, seed\-replicated drop in an expansion proxy and in long\-horizon error\. The same knob does not deliver population contraction on the action\-conditioned and natural\-video settings we tested\. A controllable stochastic\-forcing sweep places those regimes on one approximately linearL20​\(η\)L\_\{20\}\(\\eta\)curve at fixedλ=0\.8\\lambda\{=\}0\.8, and complete joint slices atλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}show the same qualitative rise\. That is enough to treatλ\\lambdaas a geometry diagnostic with known domain limits\. It is not a universal contraction law, and it is not a new world model\.

#### Practical takeaway\.

For passive\-video latents, logL20,q95L\_\{20,\\mathrm\{q95\}\}with MSE and treatλ\\lambdaas a geometry dial with a measurable threshold near11\. For action\-conditioned control, remeasureL20L\_\{20\}on that domain; do not export MMNIST’sλ=0\.8\\lambda\{=\}0\.8as a planning recipe\. Across regimes, the forcing lawL20≈a\+b​ηL\_\{20\}\\approx a\+b\\eta—checked at threeλ\\lambdavalues on the joint grid—is more portable than any single\-domain crossing event\.

## 12Open problems

These questions sit beyond the present evidence\.

#### Causal identification\.

Our mediation analysis is associational becauseλ\\lambdais a hyperparameter, not a randomized intervention\. A design that randomly assignsλ\\lambda\(or randomly injectsη\\eta\) within a fixed architecture would support stronger causal language about theλ→L→E\\lambda\\to L\\to Epath\.

#### Certified geometry\.

L20,q95L\_\{20,\\mathrm\{q95\}\}is a finite\-sample chord statistic\. Connecting it to global Lipschitz certificates\[[22](https://arxiv.org/html/2607.21645#bib.bib22),[20](https://arxiv.org/html/2607.21645#bib.bib20)\]without destroying video prediction quality remains open\.

#### Planner coupling\.

Asadi’s sweet spot appears under co\-trained model\-based control; our negative MPC result uses a frozen random\-shooting planner\. A systematic study that varies bothλ\\lambda\(orkk\) and planner coupling would clarify when geometric regularization helps returns\.

#### Richer domains\.

dm\_control cartpole\-swingup is reported as a recognized\-control sanity check in the main text \(11/1211/12cells; no populationL<1L\{<\}1\)\. Larger natural\-video corpora and stochastic world models \(RSSM\-style\) remain open tests of whether theηeff\\eta\_\{\\mathrm\{eff\}\}placement rule survives beyond Gymnasium CartPole/Pendulum and KTH\.

#### Joint\(λ,η\)\(\\lambda,\\eta\)surface\.

We report a dense primary curve atλ=0\.8\\lambda\{=\}0\.8and complete joint slices atλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}\(5​η×35\\eta\{\\times\}3seeds each\)\. Both slices track approximately linearL20​\(η\)L\_\{20\}\(\\eta\)with slopes near the primary law \(≈1\.7\\approx 1\.7–2\.02\.0\), so a coarseλ\\lambda\-dependence of the forcing slope is not evident on this grid\. A denser surface over continuousλ\\lambda\(and a formal test of slope×λ\\times\\lambdainteraction\) remains open\.

#### Architecture sensitivity\.

All main results use a GRU residual transition\. Whether the MMNISTL<1L\{<\}1threshold and theη\\eta\-law slope survive under Transformer, RSSM, or JEPA\-style predictors is unknown\. Architectural baselines in the defensive section \(antisymmetric / spectral\) are narrow substitutes, not a full architecture sweep\.

## Author contributions

Both authors contributed to experimental design, analysis, and writing\. K\.B\. led paper structure and theory exposition; A\.J\. led training pipelines, analysis scripts, and artifact packaging\.

## Reproducibility Statement

Headline statistics come from locked CSVs underresults/master/\. The package validatorreproduce\_paper\.pychecks metric JSONs and figures without retraining\. Checkpoints are omitted from the preprint bundle for size\. Core locks includeNUMBER\_LOCK\.jsonand the stage CSVs\. Training code, analysis scripts, and Kaggle kernels will be released publicly with this preprint \(the arXiv source upload contains theLaTeXpaper and figures only\)\.

## Ethics Statement

Experiments use synthetic video and standard Gymnasium simulators\. No human subjects or private data are involved\.L20L\_\{20\}must not be read as a certified stability certificate for deployment\.

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## Appendix AProof of Theorem[1](https://arxiv.org/html/2607.21645#Thmtheorem1)

We prove the bound by induction onhh\. Letztz\_\{t\}denote a reference trajectory andz^t\\hat\{z\}\_\{t\}the learned rollout, withet=‖z^t−zt‖e\_\{t\}=\\\|\\hat\{z\}\_\{t\}\-z\_\{t\}\\\|\. Assume‖z^t\+1−f​\(z^t\)‖≤ϵ\\\|\\hat\{z\}\_\{t\+1\}\-f\(\\hat\{z\}\_\{t\}\)\\\|\\leq\\epsilonand‖f​\(x\)−f​\(y\)‖≤L​‖x−y‖\\\|f\(x\)\-f\(y\)\\\|\\leq L\\\|x\-y\\\|for allx,yx,yon the rollout manifold\.

#### Base caseh=1h=1\.

e1\\displaystyle e\_\{1\}=‖z^1−z1‖=‖z^1−f​\(z0\)\+f​\(z0\)−z1‖\\displaystyle=\\\|\\hat\{z\}\_\{1\}\-z\_\{1\}\\\|=\\\|\\hat\{z\}\_\{1\}\-f\(z\_\{0\}\)\+f\(z\_\{0\}\)\-z\_\{1\}\\\|≤‖z^1−f​\(z^0\)‖\+‖f​\(z^0\)−f​\(z0\)‖≤ϵ\+L​e0\.\\displaystyle\\leq\\\|\\hat\{z\}\_\{1\}\-f\(\\hat\{z\}\_\{0\}\)\\\|\+\\\|f\(\\hat\{z\}\_\{0\}\)\-f\(z\_\{0\}\)\\\|\\leq\\epsilon\+Le\_\{0\}\.\(2\)

#### Inductive step\.

Assumeet≤Lt​e0\+ϵ​∑j=0t−1Lje\_\{t\}\\leq L^\{t\}e\_\{0\}\+\\epsilon\\sum\_\{j=0\}^\{t\-1\}L^\{j\}for somet≥1t\\geq 1\. Then the same triangle inequality as in equation[2](https://arxiv.org/html/2607.21645#A1.E2)giveset\+1≤ϵ\+L​et=ϵ\+Lt\+1​e0\+ϵ​∑j=1tLje\_\{t\+1\}\\leq\\epsilon\+Le\_\{t\}=\\epsilon\+L^\{t\+1\}e\_\{0\}\+\\epsilon\\sum\_\{j=1\}^\{t\}L^\{j\}\. Addingϵ\\epsilonto the geometric sum yieldset\+1≤Lt\+1​e0\+ϵ​∑j=0tLje\_\{t\+1\}\\leq L^\{t\+1\}e\_\{0\}\+\\epsilon\\sum\_\{j=0\}^\{t\}L^\{j\}\. Thus the claim holds fort\+1t\+1\.

#### Closed form\.

WhenL≠1L\\neq 1,∑j=0h−1Lj=\(1−Lh\)/\(1−L\)\\sum\_\{j=0\}^\{h\-1\}L^\{j\}=\(1\-L^\{h\}\)/\(1\-L\), giving the stated bound\. WhenL=1L=1, the sum equalshhandeh≤h​ϵe\_\{h\}\\leq h\\epsilon\.

#### Connection to prediction MSE\.

Ife0=0e\_\{0\}=0andϵ\\epsilonupper\-bounds one\-step model disturbance, the bound shows exponential amplification whenL\>1L\>1and saturation whenL<1L<1, matching the use ofL20,q95L\_\{20,\\mathrm\{q95\}\}as a practical monitor ofkk\-step gain rather than a certified global constant\.

## Appendix BValidity ofL20,q95L\_\{20,\\mathrm\{q95\}\}as a local expansion statistic

### B\.1Estimand and sampling protocol

Letf:𝒵→𝒵f:\\mathcal\{Z\}\\to\\mathcal\{Z\}be the \(possibly action\-conditioned\) one\-step latent transition composed intof\(k\)f^\{\(k\)\}\. For a validation distributionμ\\muon𝒵\\mathcal\{Z\}, draw pairs\(Zi,Zj\)\(Z\_\{i\},Z\_\{j\}\)i\.i\.d\. fromμ×μ\\mu\\times\\muand define

Ri​j\(k\)=‖f\(k\)​\(Zi\)−f\(k\)​\(Zj\)‖‖Zi−Zj‖,Q0\.95​\(R\(k\)\)=inf\{r:ℙ​\(Ri​j\(k\)≤r\)≥0\.95\}\.R^\{\(k\)\}\_\{ij\}=\\frac\{\\\|f^\{\(k\)\}\(Z\_\{i\}\)\-f^\{\(k\)\}\(Z\_\{j\}\)\\\|\}\{\\\|Z\_\{i\}\-Z\_\{j\}\\\|\},\\qquad Q\_\{0\.95\}\(R^\{\(k\)\}\)=\\inf\\\{r:\\mathbb\{P\}\(R^\{\(k\)\}\_\{ij\}\\leq r\)\\geq 0\.95\\\}\.Our reportedL20,q95L\_\{20,\\mathrm\{q95\}\}is the empirical9595th percentile ofnpairsn\_\{\\mathrm\{pairs\}\}ratios \(defaultnpairs=240n\_\{\\mathrm\{pairs\}\}\{=\}240,k=20k\{=\}20, stress\-split latents\), implemented without gradients inempirical\_kstep\_lipschitz\.

### B\.2Upper bound under global Lipschitzness

###### Lemma 1\(Composition bound\)\.

IfffisLL\-Lipschitz, thenf\(k\)f^\{\(k\)\}isLkL^\{k\}\-Lipschitz, and every ratio satisfiesRi​j\(k\)≤LkR^\{\(k\)\}\_\{ij\}\\leq L^\{k\}\.

###### Proof\.

Induct onkk\. Fork=1k\{=\}1,Ri​j\(k\)≤LR^\{\(k\)\}\_\{ij\}\\leq Lby definition\. If‖f\(k\)​\(x\)−f\(k\)​\(y\)‖≤Lk​‖x−y‖\\\|f^\{\(k\)\}\(x\)\-f^\{\(k\)\}\(y\)\\\|\\leq L^\{k\}\\\|x\-y\\\|, then‖f\(k\+1\)​\(x\)−f\(k\+1\)​\(y\)‖=‖f​\(f\(k\)​\(x\)\)−f​\(f\(k\)​\(y\)\)‖≤L⋅Lk​‖x−y‖\\\|f^\{\(k\+1\)\}\(x\)\-f^\{\(k\+1\)\}\(y\)\\\|=\\\|f\(f^\{\(k\)\}\(x\)\)\-f\(f^\{\(k\)\}\(y\)\)\\\|\\leq L\\cdot L^\{k\}\\\|x\-y\\\|\. ThusRi​j\(k\)≤LkR^\{\(k\)\}\_\{ij\}\\leq L^\{k\}for all pairs, andL20,q95≤L20L\_\{20,\\mathrm\{q95\}\}\\leq L^\{20\}always\. ∎

Lemma[1](https://arxiv.org/html/2607.21645#Thmlemma1)clarifies whatL20,q95<1L\_\{20,\\mathrm\{q95\}\}<1*does not*imply: it is a statement about*empirical2020\-step pair ratios*, not that the per\-step Lipschitz constantLLis below11\. It*does*imply that on95%95\\%of sampled pairs the2020\-step map is non\-expansive in the finite\-difference sense, a strictly empirical notion tied to the pair distribution\.

### B\.3Local Jacobian interpretation

WhenffisC1C^\{1\}, defineJf​\(z\)=∂f​\(z\)/∂zJ\_\{f\}\(z\)=\\partial f\(z\)/\\partial z\. For a fixed unit directionvvand smallδ\>0\\delta\>0, Taylor expansion gives

‖f\(k\)​\(z\+δ​v\)−f\(k\)​\(z\)‖δ⟶‖\(Jf​\(z\)k​v\)‖as​δ→0,\\frac\{\\\|f^\{\(k\)\}\(z\+\\delta v\)\-f^\{\(k\)\}\(z\)\\\|\}\{\\delta\}\\longrightarrow\\bigl\\\|\(J\_\{f\}\(z\)^\{k\}v\)\\bigr\\\|\\quad\\text\{as \}\\delta\\to 0,provided rollout actions \(if any\) are matched across the pair\. Thus, for small‖Zi−Zj‖\\\|Z\_\{i\}\-Z\_\{j\}\\\|, ratios track directionalkk\-step gain of the composed Jacobian along the chordZj−ZiZ\_\{j\}\-Z\_\{i\}\. The9595th percentile over*random chords*estimates a high quantile of local directional expansion on the validation manifold, notsupzρ​\(Jf​\(z\)k\)\\sup\_\{z\}\\rho\(J\_\{f\}\(z\)^\{k\}\)\.

### B\.4One\-step spectral probe

For a single latentz0z\_\{0\}, power iteration onJf​\(z0\)J\_\{f\}\(z\_\{0\}\)estimates the dominant singular direction of the*one\-step*Jacobian\. Letρ1​\(z0\)=‖Jf​\(z0\)‖2\\rho\_\{1\}\(z\_\{0\}\)=\\\|J\_\{f\}\(z\_\{0\}\)\\\|\_\{2\}\(spectral norm\)\. Under uniform smoothness,ρ1​\(z0\)k\\rho\_\{1\}\(z\_\{0\}\)^\{k\}upper\-bounds‖\(Jf​\(z0\)k​v\)‖/‖v‖\\\|\(J\_\{f\}\(z\_\{0\}\)^\{k\}v\)\\\|/\\\|v\\\|for allvv\. Hence a drop in meanρ1\\rho\_\{1\}acrossz0z\_\{0\}whenλ\\lambdaincreases is a*necessary directional indicator*that some local gains have shrunk, whileL20,q95L\_\{20,\\mathrm\{q95\}\}aggregateskk\-step finite differences over*pairs*and can be more conservative or liberal depending on chord orientation and action sampling\.

### B\.5Finite\-sample guarantee

LetQ^0\.95\\hat\{Q\}\_\{0\.95\}be the sample9595th percentile fromnni\.i\.d\. ratios\. Standard order\-statistics theory givesn​\(Q^0\.95−Q0\.95​\(R\(k\)\)\)→𝒩​\(0,σ2\)\\sqrt\{n\}\(\\hat\{Q\}\_\{0\.95\}\-Q\_\{0\.95\}\(R^\{\(k\)\}\)\)\\to\\mathcal\{N\}\(0,\\sigma^\{2\}\)under smooth distributions; in practicen=240n\{=\}240yields stable estimates across seeds but should be read with bootstrap bands whennseedsn\_\{\\mathrm\{seeds\}\}is small\.

### B\.6Coverage conditions \(when the proxy is informative\)

L20,q95L\_\{20,\\mathrm\{q95\}\}tracks local Lipschitz behavior*only if*:

1. 1\.Support\.Validation latents cover the manifold used at test rollouts \(we use a stress split\)\.
2. 2\.Chord scale\.Pairs are not dominated by near\-duplicate points \(ratios numerically unstable\) nor by extremely long chords that leave the linearization regime; our implementation clamps‖zi−zj‖≥10−8\\\|z\_\{i\}\-z\_\{j\}\\\|\\geq 10^\{\-8\}and samples uniformly over the pool\.
3. 3\.Action alignment\.For action\-conditioned rollouts, both trajectories must see the same action sequence \(random, zero, or matched protocols in Appendix[E](https://arxiv.org/html/2607.21645#A5)\); otherwise ratios mix transition geometry with action noise\.
4. 4\.Differentiability\.Jacobian interpretation requiresC1C^\{1\}transitions; ReLU/tanh GRU maps are piecewise smooth, so power\-iteration probes are local rather than global certificates\.

### B\.7Empirical alignment with Jacobian probe

Short \(six\-epoch\) MMNIST runs atλ∈\{0,0\.8\}\\lambda\\in\\\{0,0\.8\\\}give:L20,q95=1\.295→1\.069L\_\{20,\\mathrm\{q95\}\}=1\.295\\to 1\.069and mean one\-step spectral radius1\.285→1\.0181\.285\\to 1\.018\(jacobian\_proxy\.json\)\. Both statistics move in the same direction whenλ\\lambdarises, as predicted when training shrinks local Jacobian gain andkk\-step chord ratios inherit that shrinkage along most sampled directions\. This is evidence thatL20,q95L\_\{20,\\mathrm\{q95\}\}*tracks*local stability training signals; it is not a proof thatL20,q95<1L\_\{20,\\mathrm\{q95\}\}<1certifies global contractivity\.

## Appendix CHyperparameters

Table 2:Default hyperparameters \(src/config\.py\)\.#### Compute notes\.

Primary MMNIST andη\\eta\-law training ran on Kaggle GPU \(1111–1212min/job median\)\. CPU fill\-ins \(CartPole extras, DMC swingup\) use the same code paths withREQUIRE\_CUDA=0\. Wall times on CPU are typically33–8×8\\timeslonger per job\.

## Appendix DMMNIST critical\-pair seed table

Pairedn=6n\{=\}6atλ∈\{0,0\.8\}\\lambda\\in\\\{0,0\.8\\\}\(seeds\{42,…,47\}\\\{42,\\ldots,47\\\}\)\. Values areL20,q95L\_\{20,\\mathrm\{q95\}\}andE20E\_\{20\}\(stress horizon2020\)\.

Four of six seeds sit belowL=1L\{=\}1atλ=0\.8\\lambda\{=\}0\.8\(43,45,46,4743,45,46,47\)\.

## Appendix EPendulum mechanism ablations

Atλ=0\.8\\lambda\{=\}0\.8,L20L\_\{20\}under random, zero, and matched\-action protocols is approximately1\.811\.81,1\.851\.85, and1\.731\.73\(single\-checkpoint eval\)\.

## Appendix FExogenous digit table

## Appendix GScaling summary

d=16d\{=\}16:L20=0\.91±0\.43L\_\{20\}\{=\}0\.91\{\\pm\}0\.43atλ=0\\lambda\{=\}0;0\.69±0\.140\.69\{\\pm\}0\.14atλ=0\.6\\lambda\{=\}0\.6;0\.75±0\.200\.75\{\\pm\}0\.20atλ=0\.8\\lambda\{=\}0\.8\.d=32d\{=\}32:3\.47±0\.703\.47\{\\pm\}0\.70/0\.89±0\.110\.89\{\\pm\}0\.11/1\.01±0\.021\.01\{\\pm\}0\.02atλ∈\{0,0\.6,0\.8\}\\lambda\\in\\\{0,0\.6,0\.8\\\}\.d=128d\{=\}128:4\.96±3\.054\.96\{\\pm\}3\.05/1\.20±0\.051\.20\{\\pm\}0\.05/1\.14±0\.041\.14\{\\pm\}0\.04\.d=64d\{=\}64: Table[1](https://arxiv.org/html/2607.21645#S6.T1)\.

## Appendix HMediation bootstrap

Run\-resampled bootstrap \(B=2000B\{=\}2000\) on2727MMNIST runs\. Primary:r^=0\.94\\hat\{r\}\{=\}0\.94, CI\[0\.88,1\.00\]\[0\.88,1\.00\]\. Run\-level sensitivity:r^≈0\.80\\hat\{r\}\{\\approx\}0\.80, CI\[0\.65,0\.95\]\[0\.65,0\.95\]\. Source:statistical\_audit\.json\.

## Appendix IJoint\(λ,η\)\(\\lambda,\\eta\)boundary

A secondary GPU arm variedλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}with a coarserη\\etagrid and three seeds \(30/3030/30cells locked\)\. Figure[5](https://arxiv.org/html/2607.21645#A9.F5)overlays those slices on the primaryλ=0\.8\\lambda\{=\}0\.8curve\.

![Refer to caption](https://arxiv.org/html/2607.21645v1/x7.png)Figure 5:Joint\(λ,η\)\(\\lambda,\\eta\)slices atλ∈\{0\.4,1\.2\}\\lambda\\in\\\{0\.4,1\.2\\\}\(n=3n\{=\}3per cell\) against the primaryλ=0\.8\\lambda\{=\}0\.8forcing curve \(n=5n\{=\}5\)\. All three rise approximately linearly inη\\eta\.Summary by\(λ,η\)\(\\lambda,\\eta\):

Both slices rise roughly linearly inη\\eta\(descriptive fitsL^20≈1\.30\+1\.69​η\\hat\{L\}\_\{20\}\\approx 1\.30\+1\.69\\etaatλ=0\.4\\lambda\{=\}0\.4andL^20≈1\.13\+2\.00​η\\hat\{L\}\_\{20\}\\approx 1\.13\+2\.00\\etaatλ=1\.2\\lambda\{=\}1\.2\), consistent with the primaryλ=0\.8\\lambda\{=\}0\.8law\. We do not fit a continuous surface overλ\\lambdafrom two slices\.

## Appendix JStatistical procedures

#### Paired tests\.

For MMNISTλ∈\{0,0\.8\}\\lambda\\in\\\{0,0\.8\\\}we pair by seed and apply a two\-sided pairedtt\-test and a Wilcoxon signed\-rank test toL20L\_\{20\}andE20E\_\{20\}\. We treatp<0\.05p\{<\}0\.05as the reporting threshold and quote exactpp\-values in the main text\.

#### Mediation\.

Baron–Kenny path ratios use z\-scored predictors onλ\\lambda\-averaged means for the primary point estimate, with a run\-resampled bootstrap \(B=2000B\{=\}2000\) that redraws the2727runs, re\-aggregates, and recomputes the ratio\. A sensitivity analysis repeats the paths on all run\-level tuples without averaging\.

#### η\\eta\-law\.

Per\-η\\etameans usen=5n\{=\}5seeds with Studentttintervals\. The slope ofL^20=a\+b​η\\hat\{L\}\_\{20\}=a\+b\\etauses a seed\-block bootstrap: resample the five seed panels with replacement, refit, and take the2\.52\.5/97\.597\.5percentiles ofbb\. Functional\-form comparison \(linear / threshold / power\) is descriptive; we claim a law only whenR2≥0\.5R^\{2\}\\geq 0\.5and intervals exclude zero slope\.

## Appendix Ks3 noisy\-MNIST proxy

L20L\_\{20\}only \(E20E\_\{20\}not logged\),n=3n\{=\}3:λ=0\\lambda\{=\}0:1\.25±0\.091\.25\{\\pm\}0\.09;λ=0\.8\\lambda\{=\}0\.8:1\.04±0\.041\.04\{\\pm\}0\.04;λ=1\.2\\lambda\{=\}1\.2:1\.04±0\.011\.04\{\\pm\}0\.01\.

![Refer to caption](https://arxiv.org/html/2607.21645v1/x8.png)

![Refer to caption](https://arxiv.org/html/2607.21645v1/x9.png)

Figure 6:s3 proxy \(LLonly\) and Pendulum sweep\.
## Appendix LJacobian probe

Short \(6\-epoch\) MMNIST models atλ∈\{0,0\.8\}\\lambda\\in\\\{0,0\.8\\\}; mean one\-step Jacobian spectral radius via power iteration on∂zt\+1/∂zt\\partial z\_\{t\+1\}/\\partial z\_\{t\}\(scripts/jacobian\_proxy\_eval\.py\)\. Atλ=0\\lambda\{=\}0:L20,q95=1\.295L\_\{20,\\mathrm\{q95\}\}\{=\}1\.295, mean spectral radius1\.2851\.285\. Atλ=0\.8\\lambda\{=\}0\.8:L20,q95=1\.069L\_\{20,\\mathrm\{q95\}\}\{=\}1\.069, mean spectral radius1\.0181\.018\. Both drop withλ\\lambda, consistent with Appendix[B](https://arxiv.org/html/2607.21645#A2); neither certifiessup‖Jf‖<1\\sup\\\|J\_\{f\}\\\|<1\. Source:jacobian\_proxy\.json\.

## Appendix MCartPole table

CPU seed bumps\{45,46\}\\\{45,46\\\}are locked at all fourλ\\lambdavalues \(n=5n\{=\}5\)\.

## Appendix NWorldTest details

MFP 24 tasks, CD 16 tasks; behavioral score0–55\[[29](https://arxiv.org/html/2607.21645#bib.bib29)\]\. Balancedn=5n\{=\}5\(seeds\{42,…,46\}\\\{42,\\ldots,46\\\}\)\. In\-domainΔ\+0\.64\\Delta\{\+\}0\.64; grid OODΔ−0\.70\\Delta\{\-\}0\.70\.

## Appendix OCompute log

## Appendix PControl\-domainL20L\_\{20\}bootstrap intervals

Seed\-block bootstrap means forL20,q95L\_\{20,\\mathrm\{q95\}\}\(B=2000B\{=\}2000\) on merged CSVs \(CartPolen=5n\{=\}5allλ\\lambda; Pendulumn=5n\{=\}5atλ=0\\lambda\{=\}0,n=4n\{=\}4atλ=0\.4\\lambda\{=\}0\.4,n=3n\{=\}3elsewhere; KTHn=3n\{=\}3\)\.

All intervals lie above11; none of these domains show a populationL<1L\{<\}1event on the locked grids\.

## Appendix QReporting checklist for geometry diagnostics

We recommend the following minimum report when claiming a contraction or expansion event for a latent world model:

1. 1\.*Proxy definition\.*Exact formula for the expansion statistic \(hereL20,q95L\_\{20,\\mathrm\{q95\}\}\), pair count, action protocol, and validation pool\.
2. 2\.*Seed table\.*Per\-seed values at the critical operating point, not only means\.
3. 3\.*Paired tests\.*When claiming aλ\\lambdathreshold, report paired tests on the same seeds\.
4. 4\.*Domain boundary\.*At least one action\-conditioned or natural\-video negative, soL<1L\{<\}1is not over\-exported\.
5. 5\.*Planning readout \(optional but clarifying\)\.*A frozen\-checkpoint planner score, even if negative, to separate geometry from control utility\.
6. 6\.*Compute and incompleteness\.*GPU hours and which appendix tables remain partial \(DMC11/1211/12; Pendulum seed extras\)\.

Our main text and appendices are written to satisfy \(1\)–\(6\) for the MMNIST critical pair and theη\\eta\-law primary arm\.

## Appendix RExtended discussion of theL=1L\{=\}1reference line

The choice ofL=1L\{=\}1as a reference line is motivated by Theorem[1](https://arxiv.org/html/2607.21645#Thmtheorem1): under a global Lipschitz assumption,L<1L\{<\}1saturates disturbance accumulation whileL\>1L\{\>\}1permits exponential growth\. In practice we never observe a global Lipschitz certificate; we observe a high quantile of finite\-differencekk\-step ratios\. CrossingL20,q95<1L\_\{20,\\mathrm\{q95\}\}\{<\}1therefore means that*on the measured validation pool*,95%95\\%of sampled chords are non\-expansive under the2020\-step map\. It does*not*mean:

- •that every direction of the Jacobian spectrum is contractive;
- •that planning returns improve;
- •that the sameλ\\lambdawill produceL<1L\{<\}1on another domain;
- •that the model is safe for open\-loop deployment\.

We keep the reference line because it is interpretable, comparable across seeds, and aligned with the recurrence bound’s qualitative split\. Alternative references \(e\.g\., mean spectral radius=1=1, or a planner\-calibrated threshold\) are left as open measurement choices\.

## Appendix SReproducibility map

Primary artifacts live underresults/master\. Headline numbers are locked inNUMBER\_LOCK\.jsonand checked byreproduce\_paper\.py\. Training entry points live undersrc/; analysis and packaging scripts live underscripts/\. CPU fill\-in scripts write resume\-safe CSVs\. The arXiv source bundle contains the compiled paper assets; code and CSV locks are released separately with the public repository\.