torus-world

A toy world-model benchmark: a damped particle driven by a two-basin action process with a bimodal action-magnitude distribution (low- vs high-thrust rings) and temporal smoothing — occasional smooth hops between basins around a torus manifold. The benchmark measures long-horizon consistency as staying on the data manifold. Generated with quickdraw.

Observation / action

• observation_vector (6): [position (x,y,z); velocity (ẋ,ẏ,ż)]
• observation.images.fpv (256×256×3, video): the egocentric / first-person-view RGB frame, stored as MP4 (torchcodec-decoded), aligned 1:1 with the vector frames — the image modality for vision models
• action (2): (a_θ, a_φ) — angular pushes in the two torus coordinates

Action distribution

The action process is bimodal. Its per-timestep action-magnitude distribution (8 timesteps, regenerated with every dataset build) — read the two peaks for the bimodal sampler:

Splits

split	episodes	steps/ep	transitions	environment	coloring
train	256	256	65536	R=0.75, r=0.25, γ=0.3, a_max=4.0	hsv
val	64	256	16384	R=0.75, r=0.25, γ=0.3, a_max=4.0	hsv
eval_ood_horizon	64	1024	65536	R=0.75, r=0.25, γ=0.3, a_max=4.0	hsv
eval_ood_visual	64	256	16384	R=0.75, r=0.25, γ=0.3, a_max=4.0	circles
eval_ood_geometric	64	256	16384	R=1.1, r=0.08, γ=0.3, a_max=4.0	hsv
eval_ood_dynamics	64	256	16384	R=0.75, r=0.25, γ=0.5, a_max=4.0	hsv

train/val are in-distribution. Each eval_ood_* changes exactly one axis: visual (texture only), geometric (R, r only), dynamics (γ, a_max only); eval_ood_horizon is the in-distribution long-horizon eval.

Metrics (long-horizon rollout)

• manifold_distance_error = |signed_dist(p̂)| — perpendicular distance off the torus surface (0 = on-surface)
• pointwise_error = ‖p̂ − p‖ — distance to the true point at the same step
• tangent_velocity_error = |⟨ṗ̂, n̂⟩| — predicted velocity's component off the surface (0 = tangent)

Normalization statistics are computed on train only and applied to every split.