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Multi-Horizon Consistency as Geometry: When Latent Dynamics Contract, and When They Do Not
Kavya Bhand, Aadi Joshi
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
This study investigates the impact of multi-horizon latent consistency (weighted by lambda) on the transition geometry of world models. Using an empirical expansion proxy (L20,q95), the authors find that increasing lambda significantly reduces expansion and prediction error on passive video tasks like Moving-MNIST, pushing the system into a near-contractive regime (L<1). However, this effect does not generalize to action-conditioned control domains (Pendulum, CartPole) or natural video (KTH), where the system remains expansive despite error reduction. The paper proposes a stochastic-forcing law to unify these regimes, suggesting that passive and active dynamics can be mapped to a single curve via calibrated noise levels.
Entities (10)
Relation Signals (7)
lambda → affects → L20,q95
confidence 95% · raising lambda from 0 to 0.8 cuts L20 from 4.96 +/- 2.01 to 1.01 +/- 0.06
lambda → affects → E20
confidence 95% · raising lambda from 0 to 0.8 ... halves E20 (0.365 to 0.177)
Multi-Horizon Latent Consistency → iscomponentof → World Models
confidence 95% · Multi-horizon latent consistency is a common training knob in video predictors and world models
Pendulum-v1 → exhibits → Expansion
confidence 92% · The same loss does not produce population L<1 on action-conditioned Pendulum-v1
Moving MNIST → exhibits → Contraction
confidence 92% · Four of six seeds cross L<1 at lambda=0.8
CartPole-v1 → exhibits → Expansion
confidence 92% · The same loss does not produce population L<1 on ... CartPole-v1
Stochastic Forcing → unifies → Control Domains
confidence 88% · A stochastic-forcing law ... unifies control domains on the same curve via calibrated eta_eff
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Abstract
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 treat lambda, the weight on multi-step latent agreement, as a diagnostic control and measure an empirical expansion proxy L20,q95 together with horizon-20 prediction error E20. On Moving-MNIST (n=6 seeds at the critical pair), raising lambda from 0 to 0.8 cuts L20 from 4.96 +/- 2.01 to 1.01 +/- 0.06 (paired t p=0.005, Wilcoxon p=0.031) and halves E20 (0.365 to 0.177, paired t p=1.1e-13). Four of six seeds cross L<1 at lambda=0.8. The same loss does not produce population L<1 on action-conditioned Pendulum-v1 or CartPole-v1, nor on KTH Actions video, even when E20 improves. An associational mediation analysis on MMNIST gives r-hat=0.94 (95% CI [0.88, 1.00], n=27, B=2000); lambda was 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 law L20 ~ 1.23 + 1.82 eta at lambda=0.8 (bootstrap slope CI [1.73, 1.92], R^2=0.96) unifies control domains on the same curve via calibrated eta_eff. Complete joint slices at lambda in {0.4, 1.2} (30/30 cells, 5 eta x 3 seeds) show comparable linear L20(eta) slopes (~1.69 and ~2.00); we do not fit a continuous (lambda, eta) surface. We do not report DreamerV3 or TD-MPC2 returns.
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- Source: https://arxiv.org/abs/2607.21645v1
- Canonical: https://arxiv.org/abs/2607.21645v1
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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.com Aadi 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 treat λ, the weight on multi-step latent agreement, as a diagnostic control and measure an empirical expansion proxy L20,q95L_20,q95 together with horizon-2020 prediction error E20E_20. On Moving-MNIST (n=6n=6 seeds at the critical pair), raising λ from 0 to 0.80.8 cuts L20L_20 from 4.96±2.014.96±2.01 to 1.01±0.061.01±0.06 (paired t p=0.005p=0.005, Wilcoxon p=0.031p=0.031) and halves E20E_20 (0.365→0.1770.365→0.177, paired t p=1.1×10−13p=1.1×10^-13). Four of six seeds cross L<1L<1 at λ=0.8λ=0.8. The same loss does not produce population L<1L<1 on action-conditioned Pendulum-v1 or CartPole-v1, nor on KTH Actions video, even when E20E_20 improves. An associational mediation analysis on MMNIST gives r^=0.94 r=0.94 (95% CI [0.88,1.00][0.88,1.00], n=27n=27, B=2000B=2000); λ was 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 law L^20≈1.23+1.82η L_20≈1.23+1.82η at λ=0.8λ=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 calibrated ηeff _eff. Complete joint slices at λ∈0.4,1.2λ∈\0.4,1.2\ (30/3030/30 cells, 5η×35η×3 seeds) show comparable linear L20(η)L_20(η) slopes (≈1.69≈ 1.69 and ≈2.00≈ 2.00); we do not fit a continuous (λ,η)(λ,η) 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 1 Introduction World models and joint-embedding predictors often roll out farther than their training horizon [11, 12, 14, 18]. 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∈‖z^t+k−zt+ktf‖22,=1,3,5,10,15,20,L_cons= _k \| z_t+k-z^tf_t+k \|_2^2, =\1,3,5,10,15,20\, weighted by λ against reconstruction and one-step losses, and asks whether λ moves a finite-sample expansion proxy L20,q95L_20,q95 across the reference line L=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 crosses L<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 both L20L_20 and E20E_20 as λ rises, with several seeds crossing L<1L<1 near λ=0.8λ=0.8. The same schedule on Pendulum, CartPole, and KTH tightens spectra or error without population L<1L<1. That split is the main result, and it motivates the stochastic-forcing experiment: inject controllable noise η into passive transitions and ask whether control domains sit on the same L20(η)L_20(η) curve. They do, approximately linearly, which reframes the passive/active anecdotes as placements on one law rather than isolated failures. Joint slices at two other λ values track the same qualitative rise in η, so the forcing story is not an artifact of fixing λ=0.8λ=0.8. Contributions. 1. A quantified MMNIST threshold: at λ=0.8λ=0.8, mean L20L_20 sits near 11 (4/64/6 seeds below 11), with significant paired drops in both L20L_20 and E20E_20 (n=6n=6). 2. A negative transfer result for action-conditioned Gymnasium domains and for KTH: L<1L<1 does not appear at the population level. 3. Associational mediation (r^=0.94 r=0.94) with an explicit non-randomization caveat, plus run-level sensitivity (r^≈0.80 r≈0.80). 4. A stochastic-forcing law at fixed λ=0.8λ=0.8: L^20≈a+bη L_20≈ a+bη with bootstrap CIs on b, placing Pendulum/CartPole via calibrated ηeff _eff. 5. A joint (λ,η)(λ,η) boundary at λ∈0.4,1.2λ∈\0.4,1.2\ (30/3030/30 cells): both slices rise approximately linearly in η with slopes near the primary law (descriptive; no formal λ×ηλ×η interaction test). 6. Defensive experiments with locked multi-seed numbers: WorldTest (n=5n=5), MPC (n=3n=3, real λ=0λ=0), and scaling d∈16,32,64,128d∈\16,32,64,128\. 2 Related 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. 2.1 RSSM, 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, 13, 14, 11]. 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]. Contrastive and DeepMDP objectives also learn latent transitions for control [17, 10]. These systems implicitly care about long-horizon behavior (policies are trained on multi-step rollouts) but they rarely report an explicit expansion statistic such as L20,q95L_20,q95. Our contribution is orthogonal: we hold architecture fixed (GRU residual) and ask what a standalone multi-horizon consistency weight λ does to a measurable local expansion proxy, without claiming a new agent or benchmark score. 2.2 JEPA and representation-space prediction Joint-embedding predictive architectures predict in representation space rather than pixel space [18, 3, 6]. V-JEPA 2 extends the family to action-conditioned video [4]. JEPAs stabilize training with architectural and objective choices (masking, EMA targets, multi-crop views) that differ from our explicit ℒconsL_cons term. Pendharkar [24] 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 (reporting L20L_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. 2.3 Lipschitz bounds, contraction, and stability Hard stability constraints (spectral normalization, antisymmetric transitions, stable RNN analyses) bound gain directly [21, 8, 20, 22]. Asadi et al. [2] 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 clip k on CartPole/Pendulum models maximizes returns when policies are co-trained inside the learned model (see §9.2). 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\|_Lip. The textbook horizon-error recurrence (Theorem 1) motivates monitoring whether an estimated proxy crosses L=1L=1; Proposition 1 states when L20,q95L_20,q95 is a valid finite-sample statistic for local k-step gain (Appendix B). Concurrent theory lines pursue contractivity by other means: multi-token prediction gradients [19], contractive global memory in video stacks [16], Lieb–Robinson locality bounds on pixel dynamics [1], and variable-length latent supervision for planning [30]. 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). 2.4 Post-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]. Ruan et al. [25] 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]; 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 vary λ during training and track L20,q95L_20,q95 and E20E_20, producing a training-time geometry curve. The axes are complementary: ATM/Ruan-style checks can screen a checkpoint; a λ sweep explains how the checkpoint got there. Neither substitutes for the other. 2.5 Positioning This paper is a mechanistic diagnostic study, not a new world model. It contributes (i) a quantified passive-video threshold in L20,q95L_20,q95, (i) negative transfer on action-conditioned and natural-video domains, (i) associational mediation with explicit non-randomization caveats, (iv) a stochastic-forcing law linking passive and active regimes at fixed λ, (v) complete joint (λ,η)(λ,η) slices at λ∈0.4,1.2λ∈\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 crosses L<1L<1 while active control does not, is developed in §9.1 and tested via η in §6.5. 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 that L<1L<1 improves planning returns—on Pendulum, our MPC result argues the opposite. It does not identify a causal λ→L→Eλ→ L→ E effect (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. 3 Preliminaries Latent transition. Let zt∈⊆ℝdz_t ^d denote an encoded latent at time t. A one-step transition map f:→f:Z (optionally conditioned on action ata_t) induces the k-step map f(k)f^(k). Teacher-forced latents zt+ktfz^tf_t+k come from encoding observed frames or states; free-run predictions z^t+k z_t+k roll out f from ztz_t. Regimes. Passive video trains f without actions (Moving-MNIST, KTH). Action-conditioned control trains f(zt,at)f(z_t,a_t) on offline Gymnasium trajectories (Pendulum-v1, CartPole-v1). The same consistency weight λ is 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, Rij(k)=‖f(k)(zi)−f(k)(zj)‖zi−zj‖,Lk,q95=Q^0.95(Rij(k)).R^(k)_ij= \|f^(k)(z_i)-f^(k)(z_j)\|\|z_i-z_j\|, L_k,q95= Q_0.95 (\R^(k)_ij\ ). We report L20,q95L_20,q95 (written L20L_20 in tables) and horizon-2020 prediction MSE E20E_20. L20,q95<1L_20,q95<1 means that on 95%95\% of sampled pairs the 2020-step map is finite-difference non-expansive; it is not a certificate that sup‖Jf‖<1 \|J_f\|<1 (Appendix B). Stochastic forcing. To interpolate passive and active geometry we inject z←f(z)+ησ⋆ε/dz← f(z)+η\,σ / d, ε∼(0,I) (0,I), during training at fixed λ, and place control domains by calibrated ηeff _eff (Section 6.5). 4 Method 4.1 Models Passive video. Convolutional encoder, GRU residual transition z^t+1=zt+αtanh(Δzt) z_t+1=z_t+α ( z_t) with α=0.5α=0.5, d=64d=64, hidden width 160160 [26]. 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]. 4.2 Loss and metrics Total loss ℒ=ℒrecon+ℒ1step+λℒconsL=L_recon+L_1step+λ\,L_cons. Sweeps use λ∈0,0.2,…,1.2λ∈\0,0.2,…,1.2\ unless noted. For held-out latent pairs the model rolls out K=20K=20 steps and forms ratios rij(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,q95 is the 9595th percentile of those ratios. This is a finite-sample local expansion proxy, not a certified global Lipschitz constant [20, 22]. E20E_20 is MSE on stress-split frames at horizon 2020. 4.3 Training Default: AdamW, lr 2×10−32×10^-3, 2020 MMNIST epochs / 2525 Pendulum epochs, seeds 42,43,44\42,43,44\, with two extra MMNIST seeds at λ∈0,0.8λ∈\0,0.8\ for paired tests (n=6n=6). Hyperparameters are listed in Appendix C. 4.4 Evaluation protocol Unless noted otherwise, each (λ,seed)(λ,seed) cell is an independent training run from the same data split. L20,q95L_20,q95 uses npairs=240n_pairs=240 random chords from a held-out stress or validation pool with a small number of loader batches (default 44–2525 depending on stage; see Appendix C). Action-conditioned L20L_20 uses matched random action sequences for both members of a pair unless an ablation specifies zero or matched logged actions. We report mean± across seeds and, for the MMNIST critical pair, paired t and Wilcoxon tests. Bootstrap confidence intervals for mediation and for the η-law slope use B=2000B=2000 resamples (run-level or seed-block, respectively). No hyperparameter search over λ is claimed beyond the reported grid; λ=0.8λ=0.8 is selected as the MMNIST operating point where mean L20L_20 first sits near 11 with multiple seeds below 11. 4.5 Experimental design summary The study is organized as a sequence of locked stages rather than an open-ended leaderboard chase: 1. Primary geometry curve on Moving-MNIST (λ grid; critical pair n=6n=6). 2. Transfer negatives on Pendulum, CartPole, and KTH under the same λ grid. 3. Mechanism via stochastic forcing η at fixed λ=0.8λ=0.8 (primary n=50n=50). 4. Defenses: architectural baselines, exogenous digit, WorldTest-style scoring, latent MPC, latent-dimension scaling. 5. Joint (λ,η)(λ,η) boundary at λ∈0.4,1.2λ∈\0.4,1.2\ (30/3030/30 cells; Appendix I). Every claim in the abstract is backed by a locked CSV under results/master/ and checked by the package validator (Appendix S). Algorithm 1 Train and evaluate a consistency-weighted latent world model 1:Dataset D, weight λ, horizons K, noise level η (default 0) 2:Initialize encoder, transition f, decoder 3:for epoch =1,…,E=1,…,E do 4: for minibatch B⊂B do 5: Encode teacher-forced latents zt\z_t\; optionally add noise of scale ησ⋆ησ 6: Compute ℒrecon+ℒ1step+λ∑k∈‖z^t+k−zt+ktf‖22L_recon+L_1step+λ _k \| z_t+k-z^tf_t+k\|_2^2 7: AdamW step with gradient clipping 8: end for 9:end for 10:Encode validation pool; sample npairsn_pairs chords; compute L20,q95L_20,q95 and E20E_20 11:return checkpoint and metrics 5 Theoretical background We use two theoretical pieces: a horizon-error recurrence under Lipschitz dynamics (standard, but proved in full in Appendix A), and a validity argument for the finite-sample statistic L20,q95L_20,q95 (Appendix B). 5.1 Recurrence bound Let f:→f:Z be a latent transition map and z^t+1=f(z^t)+ξt z_t+1=f( z_t)+ _t a learned rollout with per-step disturbance ‖ξt‖≤ϵ\| _t\|≤ε. Write et=‖z^t−zt‖e_t=\| z_t-z_t\| for rollout error against a reference trajectory. Theorem 1 (Horizon error recurrence). If f is L-Lipschitz on the rollout manifold and e0=0e_0=0, then for horizon h≥1h≥ 1, eh≤Lhe0+ϵ1−Lh1−L(L≠1),eh≤hϵ(L=1).e_h≤ L^he_0+ε\, 1-L^h1-L (L≠ 1), e_h≤ hε (L=1). (1) In particular, if L<1L<1 the disturbance term saturates as h→∞h→∞; if L>1L>1, error can grow exponentially. The bound is classical in spirit [2, 5, 27]; Appendix A gives a step-by-step proof by induction. We use it only to motivate monitoring an estimated expansion statistic: crossing a proxy below 11 is suggestive of non-explosive k-step gain on the measured set, not a certificate that sup‖Jf‖<1 \|J_f\|<1. 5.2 What L20,q95L_20,q95 estimates For held-out latents ziz_i and k-step map f(k)f^(k), define pair ratios rij(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 metric L20,q95L_20,q95 is the 9595th percentile of rij(20)\r_ij^(20)\ over npairsn_pairs random pairs from validation latents (implementation: src/metrics.py). Proposition 1 (Proxy validity (informal)). Suppose f is L-Lipschitz on the support of the validation latent distribution. Then every sampled ratio satisfies rij(k)≤Lkr_ij^(k)≤ L^k. If additionally f is C1C^1, pairs are drawn with ‖zi−zj‖≤δ\|z_i-z_j\|≤δ for small δ, and rollout actions are fixed across the pair (passive video or matched-action protocol), then rij(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\| as δ→0δ→ 0. The empirical 9595th percentile is a consistent estimator of the 9595th quantile of rij(k)r_ij^(k) under i.i.d. pair resampling as npairs→∞n_pairs→∞. Appendix B states coverage conditions, relates k-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) shows both L20,q95L_20,q95 and mean one-step spectral radius drop when λ rises (1.29→1.071.29→ 1.07 and 1.29→1.021.29→ 1.02 at six epochs), consistent with Proposition 1 but not certifying a global Lipschitz constant. 6 Results 6.1 Moving-MNIST: a rapid λ threshold Table 1 and Figure 1 summarize 2727 MMNIST runs, with n=6n=6 at λ∈0,0.8λ∈\0,0.8\. L20L_20 falls from 4.96±2.014.96±2.01 to 1.01±0.061.01±0.06 (paired t p=0.005p=0.005; Wilcoxon p=0.031p=0.031). E20E_20 falls from 0.365±0.00070.365±0.0007 to 0.177±0.00030.177±0.0003 (paired t p=1.1×10−13p=1.1×10^-13; Wilcoxon p=0.031p=0.031). Seeds 43,45,46,4743,45,46,47 sit below L=1L=1 at λ=0.8λ=0.8 (4/64/6). Mean L20L_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 measured L20L_20 and E20E_20 as λ increases. No non-analytic statistical-mechanics phase transition is claimed. Table 1: Moving-MNIST λ-sweep (mean ± std). n=6n=6 at λ∈0,0.8λ∈\0,0.8\; otherwise n=3n=3. λ L20,q95L_20,q95 E20E_20 frac. L<1L<1 0.0 4.96±2.014.96± 2.01 0.365±0.00070.365± 0.0007 0/6 0.2 1.04±0.031.04± 0.03 0.177±0.00030.177± 0.0003 0/3 0.4 1.06±0.031.06± 0.03 0.177±0.00010.177± 0.0001 0/3 0.6 1.04±0.041.04± 0.04 0.177±0.00020.177± 0.0002 0/3 0.8 1.01±0.061.01± 0.06 0.177±0.00030.177± 0.0003 4/6 1.0 1.06±0.061.06± 0.06 0.177±0.00030.177± 0.0003 0/3 1.2 1.12±0.041.12± 0.04 0.177±0.00040.177± 0.0004 0/3 Figure 1: Left: MMNIST L20L_20 and E20E_20 vs. λ. Right: passive MMNIST vs. action-conditioned Pendulum (mean ± std; dashed line L=1L=1). 6.2 Pendulum and CartPole: tightening without L<1L<1 On Pendulum-v1, L20L_20 decreases from 2.51±0.172.51±0.17 at λ=0λ=0 (n=5n=5) to 1.74±0.131.74±0.13 at λ=0.8λ=0.8 (n=3n=3), then stays near 1.781.78 at λ=1.2λ=1.2. No seed crosses L<1L<1. E20E_20 is already near 10−410^-4, so error is a weak phase detector. Action-protocol ablations at λ=0.8λ=0.8 (random / zero / matched actions) leave L20L_20 in 1.73,1.81,1.85\1.73,1.81,1.85\ (Appendix E). CartPole-v1 shows the same qualitative split: L20L_20 stays expansive at every λ with n=5n=5 (2.96±0.262.96±0.26, 3.18±0.243.18±0.24, 3.17±0.423.17±0.42, 2.99±0.202.99±0.20 at λ∈0,0.4,0.8,1.2λ∈\0,0.4,0.8,1.2\) while E20E_20 drops once λ≥0.4λ≥0.4 (Appendix M). On dm_control cartpole-swingup (offline random trajectories, same GRU residual backbone), a CPU sweep yields λ=0λ=0: L20=1.41±0.01L_20=1.41±0.01 (n=3n=3); λ=0.4λ=0.4: 1.50±0.011.50±0.01 (n=3n=3); λ=0.8λ=0.8: 1.55±0.131.55±0.13 (n=3n=3); λ=1.2λ=1.2: 1.55±0.061.55±0.06 (n=2n=2; seed 4444 not completed). No population L<1L<1 appears on this grid, matching Gymnasium CartPole-v1 qualitatively (tighter absolute L20L_20, still expansive). We treat this as a recognized-control sanity check rather than a Dreamer-scale benchmark. 6.3 KTH Actions On real KTH clips (n=3n=3, seq_len 1616), L20L_20 tightens from 3.32±2.573.32±2.57 at λ=0λ=0 to 1.54±0.661.54±0.66 at λ=1.2λ=1.2. Every seed decreases, seed 4444 remains an expansive outlier, and no seed crosses L<1L<1. E20E_20 stays near 0.050.05. A noisy three-digit MNIST proxy (Appendix K) is retained only as a stress test; KTH is the natural-video result. 6.4 Latent dimension For d∈16,32,128d∈\16,32,128\ we sweep λ∈0,0.6,0.8λ∈\0,0.6,0.8\ with n=3n=3; d=64d=64 is the main MMNIST table. At d=16d=16, mean L20=0.91L_20=0.91 already at λ=0λ=0 (2/32/3 seeds L<1L<1). At d=32d=32, L20L_20 falls 3.47→0.893.47→0.89 by λ=0.6λ=0.6 (1.011.01 at λ=0.8λ=0.8). At d=128d=128, L20L_20 falls 4.96→1.144.96→1.14 and never crosses L<1L<1. Larger latents need stronger λ to approach the contractive band. This finite-size pattern is consistent with an under-constrained residual map in high d: soft multi-horizon agreement must fight more directions before the 9595th percentile of chord ratios drops below 11. 6.5 Stochastic 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)+ησ⋆ε/dz← f(z)+η\,σ / d with ε∼(0,I) (0,I), fixed λ=0.8λ=0.8, and σ⋆=0.5σ =0.5. Primary GPU sweep: η∈0,0.1,…,1.2η∈\0,0.1,…,1.2\ (10 points) × 5 seeds (n=50n=50 runs, 16 epochs). Mean L20,q95L_20,q95 increases approximately linearly, L^20≈1.23+1.82η L_20≈ 1.23+1.82η (block-bootstrap 95% CI on slope [1.73,1.92][1.73,1.92], R2=0.96R^2=0.96; Figure 2). Figure 2: Stochastic forcing at λ=0.8λ=0.8: mean L20,q95L_20,q95 vs. η on MMNIST (n=5n=5 seeds per point; SEM bars). Dotted line: least-squares fit. Vertical dotted lines: calibrated ηeff _eff for Pendulum, CartPole, and KTH. We do not observe a mean-level L=1L=1 upcrossing on this grid (L^20(η=0)=1.10 L_20(η=0)=1.10, per-η 95% CI [0.95,1.26][0.95,1.26]), but calibrated effective noise levels place Pendulum (ηeff≈0.53 _eff≈0.53) and CartPole (ηeff≈1.0 _eff≈1.0) at predicted L^20≈2.2 L_20≈ 2.2 and 3.13.1, consistent with population L>1L>1 in those domains. KTH is the exception under the same calibration: ηeff≈0 _eff≈0 by construction (no actions), yet L20L_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=5 seeds per η) are reported in eta_law_bootstrap.json. Joint boundary slices at λ∈0.4,1.2λ∈\0.4,1.2\ are complete (30/3030/30 cells; Appendix I) and show comparable linear L20(η)L_20(η) slopes; we do not claim a denser continuous (λ,η)(λ,η) surface or a finer η grid on the primary curve. 7 Associational mediation We fit Baron–Kenny paths on z-scored λ, logL20 L_20, and logE20 E_20 with DAG λ→L→Eλ→ L→ E and a direct λ→Eλ→ E edge [7, 23]. Figure 3: Associational mediation DAG used for the path-ratio analysis. Solid arrows: λ→L20→E20λ→ L_20→ E_20; dashed: direct λ→E20λ→ E_20. Point estimates use seven λ-averaged means. Confidence intervals use run-resampled bootstrap (B=2000B=2000): resample 2727 runs, re-aggregate, recompute paths. Primary result: r^=0.94 r=0.94, 95% CI [0.88,1.00][0.88,1.00]. Run-level sensitivity on all 2727 tuples: r^≈0.80 r≈0.80, CI [0.65,0.95][0.65,0.95]. Because λ is 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 that λ associates with geometry and that geometry associates with long-horizon error on MMNIST. It is not a license to say “raise λ to contract L, then E falls.” Randomizing λ (or η) inside a fixed architecture would be the next design step for causal language (Section 12). 8 Defensive experiments 8.1 What 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, 15, 29]. Those omissions are intentional. The paper is a diagnostic study of λ. Comparing against agent returns would change the estimand from geometry to control performance and would require a different experimental budget. 8.2 Architectural baselines At λ=0.8λ=0.8 on MMNIST (n=3n=3), soft consistency (L20=1.04±0.08L_20=1.04±0.08) matches spectral normalization (1.05±0.161.05±0.16) and beats AntisymmetricRNN (5.22±0.385.22±0.38), which over-constrains the dynamics (Figure 4, 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 that λ is not a strictly weaker knob than spectral normalization for the L20L_20 proxy here. 8.3 Exogenous digit Following the spirit of Pendharkar [24], a digit with resampled random velocity is added and only L20L_20/E20E_20 are tracked. At λ=1.2λ=1.2, exogenous L20=0.97±0.05L_20=0.97±0.05 and control L20=1.17±0.20L_20=1.17±0.20 (n=3n=3). No MSE/L collapse appears. That is not a refutation of exogenous discard in JEPAs; feature-retention probes are absent. 8.4 WorldTest-style scoring In-domain MMNIST behavioral scores (n=5n=5 both arms): 0.41±0.080.41±0.08 at λ=0λ=0 to 1.06±0.091.06±0.09 at λ=0.8λ=0.8 (paired Δ+0.64 +0.64). Masked-frame prediction stays near chance; change detection drives the gain. On a grid-world pixel adapter (OOD), scores move 1.88±0.35→1.17±0.521.88±0.35→ 1.17±0.52 (Δ−0.70 -0.70): consistency helps in-domain CD and hurts the OOD adapter. AutumnBench leaderboard evidence is not claimed. 8.5 Latent MPC (negative sweet-spot) Random-shooting MPC on Pendulum (H=10H=10, K=32K=32, 2020 episodes, seeds 42,43,44\42,43,44\) with real λ=0λ=0 checkpoints: mean returns −713±62-713±62 (λ=0λ=0), −756±12-756±12 (λ=0.8λ=0.8), −768±23-768±23 (λ=1.2λ=1.2). Best mean at λ=0λ=0; λ=0.8λ=0.8 wins on 1/31/3 seeds. There is no planning sweet spot at the MMNIST λ threshold. That ranking differs from an earlier single-seed run that used a λ=0.2λ=0.2 proxy for the expansive regime; the multi-seed result with true λ=0λ=0 supersedes it. See §9.2 for reconciliation with Asadi et al. [2]. The negative MPC result is load-bearing for the paper’s framing. If L<1L<1 on passive video were a sufficient condition for better latent planning on control, we would expect λ=0.8λ=0.8 to help on Pendulum. It does not under this planner. That keeps the diagnostic honest: geometry and planning utility are separable claims. Figure 4: Left: architectural baselines at λ=0.8λ=0.8. Right: Pendulum latent-MPC returns (n=3n=3 seeds, 2020 episodes each). 9 Discussion The usable claim is narrow: soft consistency contracts passive-video geometry under the L20L_20 proxy, and that effect does not automatically transfer to action-conditioned or natural-video settings. 9.1 Passive 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. Raising λ sharply moves L20,q95L_20,q95 toward 11 and cuts E20E_20, four of six seeds cross L<1L<1 at λ=0.8λ=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 same λ tightens spectra or error without population L<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 local k-step gain toward a non-expansive band (Proposition 1). 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 the 9595th percentile of k-step ratios below 11. Pendharkar [24] 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. 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. Loss competition. ℒconsL_cons can dominate passive training at high λ; on Pendulum/CartPole, near-zero E20E_20 already decouples error from L20L_20, so geometry and MSE need not move together. 3. Action routing. Matched-action ablations leave L20≈1.7L_20≈ 1.7–1.91.9 at λ=0.8λ=0.8 (Appendix E), suggesting action-conditioned Jacobians resist the same isotropic shrinkage that passive video admits. Section 6.5 tests (i)–(i) via controllable η: a single linear law with bootstrap confidence bands places passive video near η=0η=0 and control domains at higher ηeff _eff, reducing four disconnected anecdotes to one reported scaling relation. 9.2 Reconciling planning sweet spots with Asadi et al. [2] Asadi et al. [2] bound multi-step Wasserstein error and value error in terms of a model Lipschitz constant, then empirically cap weight norms by a hard bound k while learning CartPole (15×10315×10^3 tuples) and Pendulum (10410^4 tuples) transition models. For each k, 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 intermediate k maximizing average return, a planning sweet spot on spectral gain during coupled model–policy learning. Our Pendulum latent-MPC experiment is deliberately decoupled: λ is 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=3 seeds). Mean returns are best at λ=0λ=0 (−713±62-713±62) and worsen at λ=0.8λ=0.8 and 1.21.2 (−756±12-756±12, −768±23-768±23); there is no sweet spot at the MMNIST contraction threshold. These results are compatible rather than contradictory: • Different knobs: Asadi control k, a hard per-layer Lipschitz cap on the transition network; we control λ, 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 proxy L20,q95L_20,q95 and a crude MPC score. Improved geometry on MMNIST does not license transferring λ=0.8λ=0.8 to Pendulum planning. Takeaway for practitioners: spectral regularization can help planning when the regularizer, model, and planner are co-designed; a training-time λ that crosses L<1L<1 on passive video is neither necessary nor sufficient for latent MPC gains on the control domains tested here. 9.3 Diagnostics, mediation, and open questions ATM and future-compatibility diagnostics [9, 25] screen trained models for action-consistent rollouts; concurrent theory work enforces or analyzes contractivity via multi-token losses, memory gating, or locality bounds [19, 16, 1, 30]. Our λ sweep 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 with L20,q95L_20,q95. Associational mediation (r^=0.94 r=0.94 on MMNIST) supports a coherent story: λ associates with lower L20L_20 and lower E20E_20, but λ was not randomized, so causal language is avoided. The passive/active split is partly unified by the stochastic-forcing law (§6.5): soft consistency is a domain-limited geometry diagnostic—useful on passive video at low ηeff _eff, and misleading if exported wholesale to control without remeasurement on the same curve. Joint slices at λ∈0.4,1.2λ∈\0.4,1.2\ (Appendix I) track the same qualitative rise in η, which argues against treating the primary λ=0.8λ=0.8 slope as a one-off. 10 Limitations Sample sizes are modest: n=6n=6 at the critical MMNIST pair, n=5n=5 on CartPole (all λ) and on Pendulum at λ=0λ=0, n=4n=4 on Pendulum at λ=0.4λ=0.4, n=3n=3 elsewhere unless noted, and n=5n=5 seeds on the primary η grid. For the MMNIST paired L20L_20 drop (4.96→1.014.96→ 1.01), the observed effect is large relative to within-seed variability (paired t p=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,q95 is a local proxy whose validity conditions are in Appendix B; it is not sup‖Jf‖ \|J_f\|. Mediation is associational. Benchmarks are Moving-MNIST, Pendulum-v1, CartPole-v1, KTH, and dm_control cartpole-swingup (CPU sweep: 11/1211/12 cells locked; λ=1.2λ=1.2 missing seed 4444). 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)(λ,η) boundary is complete at λ∈0.4,1.2λ∈\0.4,1.2\ (n=3n=3 per cell; Appendix I): both slices rise approximately linearly in η, with slopes comparable to the primary λ=0.8λ=0.8 law; we still do not fit a full continuous surface over λ. Logged compute is on the order of 5050 GPU-hours plus multi-day CPU fill-ins (Appendix compute table). 10.1 Threats to validity Internal. λ is a training hyperparameter, not a randomized intervention, so λ→L→Eλ→ L→ E paths are associational. L20,q95L_20,q95 depends on pair sampling and validation pools; Appendix B 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<1 is a finite-difference non-expansiveness event, not a planning-quality certificate; our MPC negative result makes that distinction operational. 11 Conclusion 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 linear L20(η)L_20(η) curve at fixed λ=0.8λ=0.8, and complete joint slices at λ∈0.4,1.2λ∈\0.4,1.2\ show the same qualitative rise. That is enough to treat λ as 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, log L20,q95L_20,q95 with MSE and treat λ as a geometry dial with a measurable threshold near 11. For action-conditioned control, remeasure L20L_20 on that domain; do not export MMNIST’s λ=0.8λ=0.8 as a planning recipe. Across regimes, the forcing law L20≈a+bηL_20≈ a+bη—checked at three λ values on the joint grid—is more portable than any single-domain crossing event. 12 Open problems These questions sit beyond the present evidence. Causal identification. Our mediation analysis is associational because λ is a hyperparameter, not a randomized intervention. A design that randomly assigns λ (or randomly injects η) within a fixed architecture would support stronger causal language about the λ→L→Eλ→ L→ E path. Certified geometry. L20,q95L_20,q95 is a finite-sample chord statistic. Connecting it to global Lipschitz certificates [22, 20] 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 both λ (or k) 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 (11/1211/12 cells; no population L<1L<1). Larger natural-video corpora and stochastic world models (RSSM-style) remain open tests of whether the ηeff _eff placement rule survives beyond Gymnasium CartPole/Pendulum and KTH. Joint (λ,η)(λ,η) surface. We report a dense primary curve at λ=0.8λ=0.8 and complete joint slices at λ∈0.4,1.2λ∈\0.4,1.2\ (5η×35η×3 seeds each). Both slices track approximately linear L20(η)L_20(η) with slopes near the primary law (≈1.7≈ 1.7–2.02.0), so a coarse λ-dependence of the forcing slope is not evident on this grid. A denser surface over continuous λ (and a formal test of slope×λ×λ interaction) remains open. Architecture sensitivity. All main results use a GRU residual transition. Whether the MMNIST L<1L<1 threshold and the η-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 under results/master/. 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Let ztz_t denote a reference trajectory and z^t z_t the learned rollout, with et=‖z^t−zt‖e_t=\| z_t-z_t\|. Assume ‖z^t+1−f(z^t)‖≤ϵ\| z_t+1-f( z_t)\|≤ε and ‖f(x)−f(y)‖≤L‖x−y‖\|f(x)-f(y)\|≤ L\|x-y\| for all x,yx,y on the rollout manifold. Base case h=1h=1. e1 e_1 =‖z^1−z1‖=‖z^1−f(z0)+f(z0)−z1‖ =\| z_1-z_1\|=\| z_1-f(z_0)+f(z_0)-z_1\| ≤‖z^1−f(z^0)‖+‖f(z^0)−f(z0)‖≤ϵ+Le0. ≤\| z_1-f( z_0)\|+\|f( z_0)-f(z_0)\|≤ε+Le_0. (2) Inductive step. Assume et≤Lte0+ϵ∑j=0t−1Lje_t≤ L^te_0+ε _j=0^t-1L^j for some t≥1t≥ 1. Then the same triangle inequality as in equation 2 gives et+1≤ϵ+Let=ϵ+Lt+1e0+ϵ∑j=1tLje_t+1≤ε+Le_t=ε+L^t+1e_0+ε _j=1^tL^j. Adding ϵε to the geometric sum yields et+1≤Lt+1e0+ϵ∑j=0tLje_t+1≤ L^t+1e_0+ε _j=0^tL^j. Thus the claim holds for t+1t+1. Closed form. When L≠1L≠ 1, ∑j=0h−1Lj=(1−Lh)/(1−L) _j=0^h-1L^j=(1-L^h)/(1-L), giving the stated bound. When L=1L=1, the sum equals h and eh≤hϵe_h≤ hε. Connection to prediction MSE. If e0=0e_0=0 and ϵε upper-bounds one-step model disturbance, the bound shows exponential amplification when L>1L>1 and saturation when L<1L<1, matching the use of L20,q95L_20,q95 as a practical monitor of k-step gain rather than a certified global constant. Appendix B Validity of L20,q95L_20,q95 as a local expansion statistic B.1 Estimand and sampling protocol Let f:→f:Z be the (possibly action-conditioned) one-step latent transition composed into f(k)f^(k). For a validation distribution μ on Z, draw pairs (Zi,Zj)(Z_i,Z_j) i.i.d. from μ×μ×μ and define Rij(k)=‖f(k)(Zi)−f(k)(Zj)‖Zi−Zj‖,Q0.95(R(k))=infr:ℙ(Rij(k)≤r)≥0.95.R^(k)_ij= \|f^(k)(Z_i)-f^(k)(Z_j)\|\|Z_i-Z_j\|, Q_0.95(R^(k))= \r:P(R^(k)_ij≤ r)≥ 0.95\. Our reported L20,q95L_20,q95 is the empirical 9595th percentile of npairsn_pairs ratios (default npairs=240n_pairs=240, k=20k=20, stress-split latents), implemented without gradients in empirical_kstep_lipschitz. B.2 Upper bound under global Lipschitzness Lemma 1 (Composition bound). If f is L-Lipschitz, then f(k)f^(k) is LkL^k-Lipschitz, and every ratio satisfies Rij(k)≤LkR^(k)_ij≤ L^k. Proof. Induct on k. For k=1k=1, Rij(k)≤LR^(k)_ij≤ L by definition. If ‖f(k)(x)−f(k)(y)‖≤Lk‖x−y‖\|f^(k)(x)-f^(k)(y)\|≤ 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))\|≤ L· L^k\|x-y\|. Thus Rij(k)≤LkR^(k)_ij≤ L^k for all pairs, and L20,q95≤L20L_20,q95≤ L^20 always. ∎ Lemma 1 clarifies what L20,q95<1L_20,q95<1 does not imply: it is a statement about empirical 2020-step pair ratios, not that the per-step Lipschitz constant L is below 11. It does imply that on 95%95\% of sampled pairs the 2020-step map is non-expansive in the finite-difference sense, a strictly empirical notion tied to the pair distribution. B.3 Local Jacobian interpretation When f is C1C^1, define Jf(z)=∂f(z)/∂zJ_f(z)=∂ f(z)/∂ z. For a fixed unit direction v and small δ>0δ>0, Taylor expansion gives ‖f(k)(z+δv)−f(k)(z)‖δ⟶‖(Jf(z)kv)‖as δ→0, \|f^(k)(z+δ v)-f^(k)(z)\|δ \|(J_f(z)^kv) \| δ→ 0, provided rollout actions (if any) are matched across the pair. Thus, for small ‖Zi−Zj‖\|Z_i-Z_j\|, ratios track directional k-step gain of the composed Jacobian along the chord Zj−ZiZ_j-Z_i. The 9595th percentile over random chords estimates a high quantile of local directional expansion on the validation manifold, not supzρ(Jf(z)k) _zρ(J_f(z)^k). B.4 One-step spectral probe For a single latent z0z_0, power iteration on Jf(z0)J_f(z_0) estimates the dominant singular direction of the one-step Jacobian. Let ρ1(z0)=‖Jf(z0)‖2 _1(z_0)=\|J_f(z_0)\|_2 (spectral norm). Under uniform smoothness, ρ1(z0)k _1(z_0)^k upper-bounds ‖(Jf(z0)kv)‖/‖v‖\|(J_f(z_0)^kv)\|/\|v\| for all v. Hence a drop in mean ρ1 _1 across z0z_0 when λ increases is a necessary directional indicator that some local gains have shrunk, while L20,q95L_20,q95 aggregates k-step finite differences over pairs and can be more conservative or liberal depending on chord orientation and action sampling. B.5 Finite-sample guarantee Let Q^0.95 Q_0.95 be the sample 9595th percentile from n i.i.d. ratios. Standard order-statistics theory gives n(Q^0.95−Q0.95(R(k)))→(0,σ2) n( Q_0.95-Q_0.95(R^(k))) (0,σ^2) under smooth distributions; in practice n=240n=240 yields stable estimates across seeds but should be read with bootstrap bands when nseedsn_seeds is small. B.6 Coverage conditions (when the proxy is informative) L20,q95L_20,q95 tracks local Lipschitz behavior only if: 1. Support. Validation latents cover the manifold used at test rollouts (we use a stress split). 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\|≥ 10^-8 and samples uniformly over the pool. 3. Action alignment. For action-conditioned rollouts, both trajectories must see the same action sequence (random, zero, or matched protocols in Appendix E); otherwise ratios mix transition geometry with action noise. 4. Differentiability. Jacobian interpretation requires C1C^1 transitions; ReLU/tanh GRU maps are piecewise smooth, so power-iteration probes are local rather than global certificates. B.7 Empirical alignment with Jacobian probe Short (six-epoch) MMNIST runs at λ∈0,0.8λ∈\0,0.8\ give: L20,q95=1.295→1.069L_20,q95=1.295→ 1.069 and mean one-step spectral radius 1.285→1.0181.285→ 1.018 (jacobian_proxy.json). Both statistics move in the same direction when λ rises, as predicted when training shrinks local Jacobian gain and k-step chord ratios inherit that shrinkage along most sampled directions. This is evidence that L20,q95L_20,q95 tracks local stability training signals; it is not a proof that L20,q95<1L_20,q95<1 certifies global contractivity. Appendix C Hyperparameters Table 2: Default hyperparameters (src/config.py). Parameter Value Latent dim d 6464 (scaling runs vary; control often 3232) Hidden dim 160160 (control often 128128) Optimizer AdamW, lr 2×10−32×10^-3, wd 10−610^-6 Grad clip 1.01.0 MMNIST epochs / batch 2020 / full train split Pendulum epochs 2525 (CPU extras: 2020) CartPole epochs 1212–1515 Consistency horizons 1,3,5,10,15,20\1,3,5,10,15,20\ Residual scale α 0.50.5 Image size / seq len 32×3232×32 / 2424 (KTH seq 1616) L20L_20 pairs / batches 240240 / 44–2525 Stochastic forcing σ⋆σ 0.50.5 (primary η sweep) Seeds (default) 42,43,4442,43,44 (+45,46,4745,46,47 where noted) Compute notes. Primary MMNIST and η-law training ran on Kaggle GPU ( 1111–1212 min/job median). CPU fill-ins (CartPole extras, DMC swingup) use the same code paths with REQUIRE_CUDA=0. Wall times on CPU are typically 33–8×8× longer per job. Appendix D MMNIST critical-pair seed table Paired n=6n=6 at λ∈0,0.8λ∈\0,0.8\ (seeds 42,…,47\42,…,47\). Values are L20,q95L_20,q95 and E20E_20 (stress horizon 2020). seed L20(λ=0)L_20(λ=0) E20(λ=0)E_20(λ=0) L20(λ=0.8)L_20(λ=0.8) E20(λ=0.8)E_20(λ=0.8) 42 2.452.45 0.3640.364 1.031.03 0.1770.177 43 6.766.76 0.3650.365 0.970.97 0.1770.177 44 5.855.85 0.3640.364 1.121.12 0.1770.177 45 3.543.54 0.3650.365 0.960.96 0.1770.177 46 3.683.68 0.3660.366 0.960.96 0.1770.177 47 7.467.46 0.3660.366 1.001.00 0.1760.176 Four of six seeds sit below L=1L=1 at λ=0.8λ=0.8 (43,45,46,4743,45,46,47). Appendix E Pendulum mechanism ablations At λ=0.8λ=0.8, L20L_20 under random, zero, and matched-action protocols is approximately 1.811.81, 1.851.85, and 1.731.73 (single-checkpoint eval). Appendix F Exogenous digit table Condition λ L20L_20 E20E_20 control 0.00.0 8.09±6.448.09± 6.44 0.3630.363 control 1.21.2 1.17±0.201.17± 0.20 0.1770.177 exogenous 0.00.0 5.16±1.045.16± 1.04 0.3590.359 exogenous 1.21.2 0.97±0.050.97± 0.05 0.1710.171 Appendix G Scaling summary d=16d=16: L20=0.91±0.43L_20=0.91±0.43 at λ=0λ=0; 0.69±0.140.69±0.14 at λ=0.6λ=0.6; 0.75±0.200.75±0.20 at λ=0.8λ=0.8. d=32d=32: 3.47±0.703.47±0.70 / 0.89±0.110.89±0.11 / 1.01±0.021.01±0.02 at λ∈0,0.6,0.8λ∈\0,0.6,0.8\. d=128d=128: 4.96±3.054.96±3.05 / 1.20±0.051.20±0.05 / 1.14±0.041.14±0.04. d=64d=64: Table 1. Appendix H Mediation bootstrap Run-resampled bootstrap (B=2000B=2000) on 2727 MMNIST runs. Primary: r^=0.94 r=0.94, CI [0.88,1.00][0.88,1.00]. Run-level sensitivity: r^≈0.80 r≈0.80, CI [0.65,0.95][0.65,0.95]. Source: statistical_audit.json. Appendix I Joint (λ,η)(λ,η) boundary A secondary GPU arm varied λ∈0.4,1.2λ∈\0.4,1.2\ with a coarser η grid and three seeds (30/3030/30 cells locked). Figure 5 overlays those slices on the primary λ=0.8λ=0.8 curve. Figure 5: Joint (λ,η)(λ,η) slices at λ∈0.4,1.2λ∈\0.4,1.2\ (n=3n=3 per cell) against the primary λ=0.8λ=0.8 forcing curve (n=5n=5). All three rise approximately linearly in η. Summary by (λ,η)(λ,η): λ / η n mean L20L_20 mean E20E_20 0.40.4 / 0 33 1.091.09 0.1770.177 0.40.4 / 0.250.25 33 1.851.85 0.1770.177 0.40.4 / 0.50.5 33 2.332.33 0.1770.177 0.40.4 / 0.750.75 33 2.642.64 0.1770.177 0.40.4 / 11 33 2.802.80 0.1760.176 1.21.2 / 0 33 1.051.05 0.1770.177 1.21.2 / 0.250.25 33 1.691.69 0.1770.177 1.21.2 / 0.50.5 33 2.212.21 0.1770.177 1.21.2 / 0.750.75 33 2.672.67 0.1760.176 1.21.2 / 11 33 3.063.06 0.1770.177 Both slices rise roughly linearly in η (descriptive fits L^20≈1.30+1.69η L_20≈ 1.30+1.69η at λ=0.4λ=0.4 and L^20≈1.13+2.00η L_20≈ 1.13+2.00η at λ=1.2λ=1.2), consistent with the primary λ=0.8λ=0.8 law. We do not fit a continuous surface over λ from two slices. Appendix J Statistical procedures Paired tests. For MMNIST λ∈0,0.8λ∈\0,0.8\ we pair by seed and apply a two-sided paired t-test and a Wilcoxon signed-rank test to L20L_20 and E20E_20. We treat p<0.05p<0.05 as the reporting threshold and quote exact p-values in the main text. Mediation. Baron–Kenny path ratios use z-scored predictors on λ-averaged means for the primary point estimate, with a run-resampled bootstrap (B=2000B=2000) that redraws the 2727 runs, re-aggregates, and recomputes the ratio. A sensitivity analysis repeats the paths on all run-level tuples without averaging. η-law. Per-η means use n=5n=5 seeds with Student t intervals. The slope of L^20=a+bη L_20=a+bη uses a seed-block bootstrap: resample the five seed panels with replacement, refit, and take the 2.52.5/97.597.5 percentiles of b. Functional-form comparison (linear / threshold / power) is descriptive; we claim a law only when R2≥0.5R^2≥ 0.5 and intervals exclude zero slope. Appendix K s3 noisy-MNIST proxy L20L_20 only (E20E_20 not logged), n=3n=3: λ=0λ=0: 1.25±0.091.25±0.09; λ=0.8λ=0.8: 1.04±0.041.04±0.04; λ=1.2λ=1.2: 1.04±0.011.04±0.01. Figure 6: s3 proxy (L only) and Pendulum sweep. Appendix L Jacobian probe Short (6-epoch) MMNIST models at λ∈0,0.8λ∈\0,0.8\; mean one-step Jacobian spectral radius via power iteration on ∂zt+1/∂zt∂ z_t+1/∂ z_t (scripts/jacobian_proxy_eval.py). At λ=0λ=0: L20,q95=1.295L_20,q95=1.295, mean spectral radius 1.2851.285. At λ=0.8λ=0.8: L20,q95=1.069L_20,q95=1.069, mean spectral radius 1.0181.018. Both drop with λ, consistent with Appendix B; neither certifies sup‖Jf‖<1 \|J_f\|<1. Source: jacobian_proxy.json. Appendix M CartPole table λ L20L_20 E20E_20 0.00.0 (n=5n=5) 2.96±0.262.96±0.26 ∼4×10−4 4×10^-4 0.40.4 (n=5n=5) 3.18±0.243.18±0.24 ∼1×10−4 1×10^-4 0.80.8 (n=5n=5) 3.17±0.423.17±0.42 ∼1×10−4 1×10^-4 1.21.2 (n=5n=5) 2.99±0.202.99±0.20 ∼1×10−4 1×10^-4 CPU seed bumps 45,46\45,46\ are locked at all four λ values (n=5n=5). Appendix N WorldTest details MFP 24 tasks, CD 16 tasks; behavioral score 0–55 [29]. Balanced n=5n=5 (seeds 42,…,46\42,…,46\). In-domain Δ+0.64 +0.64; grid OOD Δ−0.70 -0.70. Appendix O Compute log Stage Jobs Notes s1 MMNIST 2727 7λ×37λ×3 plus critical-pair extras s1 Pendulum 2121 7λ×37λ×3 s2 baselines 66 antisymmetric + spectral s2 exogenous 1212 2×2×32×2×3 s3 proxy 2121 noisy MNIST s4 scaling 2727 d∈16,32,128d∈\16,32,128\ s5 CartPole 1212 4λ×34λ×3 s7 KTH 1212 4λ×34λ×3 s8 η-law primary 5050 λ=0.8λ=0.8, 10η×510η×5 s8 η-law joint 3030 λ∈0.4,1.2λ∈\0.4,1.2\, 5η×35η×3 CPU DMC swingup 11/1211/12 cartpole-swingup; missing λ=1.2λ=1.2 seed 4444 CPU seed bump CartPole 8/88/8 seeds 4545–4646 → n=5n=5 all λ CPU seed bump Pendulum 3/83/8 λ=0λ=0 at n=5n=5; rest partial Total logged ∼200+ 200+ ∼50 50 GPU-h + CPU marathon Appendix P Control-domain L20L_20 bootstrap intervals Seed-block bootstrap means for L20,q95L_20,q95 (B=2000B=2000) on merged CSVs (CartPole n=5n=5 all λ; Pendulum n=5n=5 at λ=0λ=0, n=4n=4 at λ=0.4λ=0.4, n=3n=3 elsewhere; KTH n=3n=3). λ Pendulum CartPole KTH 0 2.512.51 [2.39,2.64]2.39,2.64] 2.962.96 [2.78,3.18]2.78,3.18] 3.323.32 [1.73,6.28]1.73,6.28] 0.40.4 1.831.83 [1.73,1.92]1.73,1.92] 3.183.18 [3.01,3.37]3.01,3.37] 1.971.97 [1.13,3.57]1.13,3.57] 0.80.8 1.741.74 [1.63,1.87]1.63,1.87] 3.173.17 [2.88,3.51]2.88,3.51] 1.411.41 [1.12,1.83]1.12,1.83] 1.21.2 1.781.78 [1.74,1.82]1.74,1.82] 2.992.99 [2.82,3.14]2.82,3.14] 1.541.54 [1.15,2.30]1.15,2.30] All intervals lie above 11; none of these domains show a population L<1L<1 event on the locked grids. Appendix Q Reporting checklist for geometry diagnostics We recommend the following minimum report when claiming a contraction or expansion event for a latent world model: 1. Proxy definition. Exact formula for the expansion statistic (here L20,q95L_20,q95), pair count, action protocol, and validation pool. 2. Seed table. Per-seed values at the critical operating point, not only means. 3. Paired tests. When claiming a λ threshold, report paired tests on the same seeds. 4. Domain boundary. At least one action-conditioned or natural-video negative, so L<1L<1 is not over-exported. 5. Planning readout (optional but clarifying). A frozen-checkpoint planner score, even if negative, to separate geometry from control utility. 6. Compute and incompleteness. GPU hours and which appendix tables remain partial (DMC 11/1211/12; Pendulum seed extras). Our main text and appendices are written to satisfy (1)–(6) for the MMNIST critical pair and the η-law primary arm. Appendix R Extended discussion of the L=1L=1 reference line The choice of L=1L=1 as a reference line is motivated by Theorem 1: under a global Lipschitz assumption, L<1L<1 saturates disturbance accumulation while L>1L>1 permits exponential growth. In practice we never observe a global Lipschitz certificate; we observe a high quantile of finite-difference k-step ratios. Crossing L20,q95<1L_20,q95<1 therefore means that on the measured validation pool, 95%95\% of sampled chords are non-expansive under the 2020-step map. It does not mean: • that every direction of the Jacobian spectrum is contractive; • that planning returns improve; • that the same λ will produce L<1L<1 on 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 S Reproducibility map Primary artifacts live under results/master. Headline numbers are locked in NUMBER_LOCK.json and checked by reproduce_paper.py. Training entry points live under src/; analysis and packaging scripts live under scripts/. 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.