How Does Geometry Enter Generated Motion?

Weihan Li; Junhao Wu; Yuhan Song; Xiaofeng Lin; Xinlei Chen

Figure 1 : Same law, different geometry. One ball, one law, one starting point; only the height of a barrier changes. In the simulated motion (top) the ball crosses the lower barrier and turns back at the one that rises above its release height (dashed line). In the generated motion (bottom; MiniMax-H3, one seed) it crosses both, as it does with every seed.

Figure 2 : Study design. Left: a world

W=(\Omega,L,x_{0})

is rendered as a first frame; the video generator produces a video and the simulator the physical motion with its tolerance band; both are compared in the coordinates of the board. Middle: two families of geometry with matched global statistics. Right: paired interventions, each changing one thing: shape ( Sec. 4.1 ); local edit, ramp pair, barrier and contact chain ( Sec. 4.2 ); prompt pair and clip length ( Sec. 4.3 ); photographed first frame and regeneration ( Sec. 4.4 ).

Table 1 : Nine models under the same law. Medians over scenes; the first row is the physical oracle, the simulator read through the same measurement chain. Intact: share of intact videos in the track and the deflector family. Gain and phase are read on tracks with 3 cycles per metre, the trend ratio on straight tracks,

e_{n}

at the first contact, the order of contacts on three deflectors, and the height ratio on ramp pairs. Tray arrival is given under the named and the neutral prompt, and

\gamma

is the exponent of tempo against clip length (

\dagger

: twice as fast at 3 s, unchanged from 5 to 8 s). A dash marks a reading that was not taken for that model.

Figure 3 : Geometry modulates the motion. (a) Gain and phase of the speed’s response to the drawn undulation; physics sits at gain 1 and phase 0. (b) The same gain against the trend ratio: the undulation is followed while the mean slope adds little. (c) Speed along a straight ramp, normalized: physics accelerates, the models hold a preferred speed. (d) Restitution at the first contact; the prescribed value is 0.85. Filled markers are open-weight models and hollow markers closed ones; grey bands are measurement floors.

Figure 4 : Cases. Five scenes (rows) in the simulated motion and in four models (columns). The ball is drawn at equal time steps, from light to full, so that its spacing shows the speed; the white dashed curve is the physical path and the blue dashed line the release height. The third row repeats the scene of the second with a prompt that leaves the outcome unnamed. Each model contributes the video closest to its median reading on that scene.

Figure 5 : Composition and locality. (a) Share of three-deflector videos with the physical order of contacts, and exit residual by contact index; the rows name the models, and a dash marks a reading that was not taken. (b) Change of speed at a bump or dip against physics, and change of the path before the ball reaches the edit, relative to an edit of equal size off the track; faded marks do not separate from the off-track edit.

Figure 6 : Energy. (a) Height reached over release height on ramp pairs, one mark per video with bars at the median; physics lies in the band below 1, and the glyphs under the swarms name the models in the order of the rows of (b). (b) Barriers, one ball per seed, solid where the ball crossed and hollow where it stopped; columns give the barrier height in percent of the release height, tinted above it. The physical oracle crosses below the release height and none above; its ball is filled by the share of the 64-member ensemble that crosses.

Figure 7 : Text controls the destination, clip length the timing. (a) Tray arrival under the prompt that names the outcome (marker) and under the neutral prompt (arrow head), for the same first frames and seeds; ticks mark the arrival the simulator reaches from the model’s own exit state, and the dashed line the simulator from the initial state. (b) Arrival time against clip length; the dashed line shows a fixed physical rate, the solid line scaling with the clip. (c) One scene and seed at 3, 5 and 8 s: position against normalized time.

Table 2 : Photographed first frames and regeneration. (a) Change of each reading from rendered to photographed first frames of the same geometry: a check where the reading stays within the seed-to-seed spread and the change where it leaves it, bold when it is attributed to the photograph and in parentheses when it lies within the range of the weaker control; a dash where there is no reading. (b) Displacement

\lambda

of the regenerated motion from the source (0) toward physics (1), by source and noise level.

ramp pairs				barriers					chutes
ascent	turn (s)	ratio	travel (mm)	ratio	outcome	time (s)	speed (m/s)	pass	$\kappa$ (m -1 )	exit, land (s)	$N_{\min}$	landing (m)
60 ∘ ∗	0.54	0.980	35 (33–37)	0.70	pass	0.78	0.51 (0.53)	64	0 ∗	0.31, 0.53	0.82	0.714–0.744
30 ∘	0.64	0.975	92 (84–96)	0.85 ∗	pass	0.83	0.34 (0.35)	64	+4	0.29, 0.51	0.82	0.714–0.742
15 ∘ ∗	0.87	0.965	183 (162–193)	0.95	pass	0.90	0.14 (0.15)	27	−4 ∗	0.34, 0.56	0.61	0.714–0.741
5 ∘	1.74	0.929	509 (411–572)	0.971	pass	1.03	0.03 (0.03)	10	+8 ∗	0.26, 0.48	0.49	0.715–0.741
0 ∘ ∗	stop 1.36	–	–	1.05 ∗	turn back	0.86	–	0

Table A.4 : Ramp pairs, barriers and chutes. Times are physical, counts are ensemble members out of 64, and ∗ marks scenes also run on the closed models.

Table A.5 : Tray interventions, near misses, permuted layouts and prompt pairs, all with the neutral prompt. Shifts are relative to the nominal landing point, positive toward the incoming ball. Counts are ensemble members out of 64. ∗ : also run on the closed models.

photograph	scene	as-built against design	ensemble	nominal band	control
	straight	−16.3/+52.3 mm, +0.07 ∘ ; $\bar{a}$ 0.093 $\to$ 0.092	stable	outside, 4% faster	twin
	$f=3$	−14.9/+42.5 mm, −1.36 ∘ ; $\bar{a}$ 0.055 $\to$ 0.076; $A$ 15 $\to$ 17.5 mm	stable	outside, 19% faster	twin
	$f=4$	−10.8/+28.6 mm, −1.03 ∘ ; $\bar{a}$ 0.048 $\to$ 0.077; $A$ 15 $\to$ 14.7 mm	stable	outside 20% of the time	twin
	broadband	−13.8/+47.7 mm, −0.62 ∘ ; $\bar{a}$ 0.069 $\to$ 0.078; $A$ 10.6/11.5/10.2 mm	stable	outside, 15% faster	twin
	single, 30 ∘	deflector 31.7 ∘ , $149\times 22$ mm; exit 0.815 m/s	1.00 / 1.00	inside	main-set render
	single, 60 ∘	deflector 61.1 ∘ , $149\times 23$ mm; exit 0.872 m/s	1.00 / 1.00	contact 17 ms late	twin

Table A.6 : Photographed first frames: as-built geometry against the design (track offset in

x/y

and rotation, launch exit speed with design 0.760 m/s), the as-built ensemble (on tracks all 64 members reach the stop, on deflectors the shares keeping the design order and resting in the tray), the nominal band, and the rendered control.

Table B.1 : The stated law (nominal) and the ensemble (64 members).

\mathcal{U}

: uniform,

\mathcal{N}

: normal. The inertia factor is that of a solid (2/5) or a hollow (2/3) ball. The bars show each uniform range (band) with the nominal value (tick) on a scale from 0 to the range’s upper end.

Table C.1 : Readings used in the main paper, as definition cards: symbol and name, definition, the physical oracle and the floor at the three native frame rates (16 / 24 / 30 fps).

Table C.2 : Fingerprints of synthetic mechanisms on tracks with

A=20

mm and

\bar{a}=0.06

at

f=2,3,4,5

m -1 (values listed in that order and drawn as a line from

f=2

to 5, with the physical value dashed: gain on 0 to 2.1, phase on

-80

to

+100^{\circ}

), 24 fps,

1280\times 720

, 1 px noise, every motion slowed

2\times

and started 1 s late. Accuracy at

f=3

, medians of three seeds.

Table C.3 : Track readings on motions with a known error (

f=3

m -1 and straight tracks, 24 fps,

832\times 480

, 1 px noise, median of three seeds). Tinted: the reading that names the error.

Table C.4 : Injected perturbations and how the integrity rule reads them: passes, flagged.

level	fps	$\|H\|$	phase	$2f/1f$	$\|b\|$	kin.	acc.
$f=3$	16	0.09	6.8 ∘	0.21	0.09	0.049	0.81
	24	0.11	7.5 ∘	0.15	0.06	0.027	0.86
	30	0.06	6.0 ∘	0.13	0.05	0.059	0.86
$f=4$	16	0.08	4.5 ∘	0.16	0.09	0.038	0.93
	24	0.07	5.6 ∘	0.15	0.08	0.026	0.90
	30	0.04	4.1 ∘	0.10	0.05	0.032	0.94

Table C.5 : Track floors at

832\times 480

: 95th percentiles of

\bigl||H|-1\bigr|

,

|

phase

|

,

2f/1f

,

|b|

and the kinematic-gain error, and the lowest balanced accuracy. On the straight level

p

reads 1.96–2.02 and

|b|\leq 0.06

s -1 .

fps	size	order	residual max	p90 max	$e_{n}$
16	$1280\times 720$	100%	4.8 ∘	2.9 ∘	0.83–0.88
	$832\times 480$	100%	8.6 ∘	4.2 ∘	0.81–0.89
24	$1280\times 720$	100%	2.8 ∘	2.1 ∘	0.82–0.89
	$832\times 480$	100%	4.3 ∘	3.2 ∘	0.68–0.91
30	$1280\times 720$	100%	3.0 ∘	1.5 ∘	0.81–0.88
	$832\times 480$	100%	4.4 ∘	2.3 ∘	0.79–0.90

Table C.6 : Deflector floors on the 24 main-set layouts (20 noise seeds per scene): exact order, largest residual, largest per-scene 90th percentile of the residual, and the range of

e_{n}

(stated law 0.85).

reading	simulator	exact renders	at tempo 2.5
local edit, bump: $dv$ / $dt$	−11.4 / +2.8%	−10.9 to −11.8 / +2.7 to +3.0%	−11.1 to −11.4 / +2.6 to +3.0%; p95 2.4–3.9 / 1.1–1.5%
local edit, dip: $dv$ / $dt$	+7.8 / −0.8%	+7.3 to +8.1 / −0.6 to −1.3%	+7.5 to +8.2 / −0.5 to −1.1%; p95 2.8–5.6 / 1.3–1.6%
local edit: $dt$ of the bump over its simulated value	1	1.07–1.10	0.94–1.06
null edit minus base: $dv$ / $dt$	0	$\leq 0.08$ / $\leq 0.03\%$	p95 3.0–4.6 / 1.0–1.6% ; flagged in 0 of 120 runs
path RMS before the edit	0	p95 0.07–0.09 mm	p95 4.2–4.8 mm
shared front: phase slope; front dispersion	1; 0	1.00; 0.03–0.05 mm	0.92–1.02 (single-video phase p95 19–32 ∘ ) ; 1.6–1.7 mm

Table C.7 : Intervention readings on exact renders (four frame-rate and size combinations) and their floors at a model-like tempo (simulated motion slowed

2.5\times

, 1 px noise, 16 / 24 / 30 fps): median over 36 pairs, and the 95th percentile (p95) of the error of a single pair or video.

	track	deflector	local edit,	ramps, barriers,	trays,	clip length
model	main set	main set	shared front	chutes	prompt pairs	3 s	8 s	photographed	regeneration
LTX-2.3	104/107 (120)	1/14 (72)	155/167 (216)	28/32 (45)	20/33 (120)	17/19 (24)	8/9 (24)	18/19 (45)	not run
LTX-2.5 (distilled)	120/120 (120)	54/57 (72)	216/216 (216)	36/39 (45)	56/67 (120)	22/22 (24)	18/20 (24)	35/39 (45)	63/68, 68/68, 58/60 (72)
LTX-2.5	99/100 (120)	4/ – (72)	209/209 (216)	20/20 (33)	not run	not run	not run	13/13 (24)	not run
Cosmos 3 Nano	106/106 (120)	48/48 (72)	174/174 (216)	33/33 (45)	49/57 (120)	not run	not run	40/40 (45)	not run
MiniMax-H3	120/120 (120)	63/68 (72)	216/216 (216)	45/45 (45)	119/119 (120)	21/21 (24)	23/23 (24)	42/42 (45)	not run

Table D.1 : Intact / readable videos per model and experiment, with the number of videos in brackets. The bars draw the intact (dark) and readable (light) shares of that number. Clip length: the new 3 s and 8 s clips (the 5 s clips are main-set videos). Photographed: photograph and twin rows together. Regeneration: the three noise levels of each model. – : not read (LTX-2.5 on deflectors).

example	form	model	where and how often
	struck deflector rotates at or after impact	Seedance 2.0, Seedance mini	deflector main set, apparatus 14/72 and 7/72; one deflector turned 26 ∘ and moved 17 mm for 92 frames
	tray tilts or shifts after the landing	LTX-2.3, MiniMax-H3	deflector main set (LTX-2.3: 13–24 mm and 6.6–8.5 ∘ ); MiniMax-H3 apparatus 9/72, mostly after the landing
	launch ramp rotates to another angle	LTX-2.5 dist.	deflector main set, 3–9 ∘ , 13 of its 18 flagged videos, 9 of them after the ball has left the ramp; trays and prompt pairs 62/120
	deflectors move as loose objects	LTX-2.3	apparatus on 71/72 deflector main-set and 100/120 tray and prompt videos; on photographs a deflector is knocked 125 mm and turned 124 ∘ , on twins it ends in the tray; chute trays move 7–27 mm in 12/12
	rails shift against the board	LTX-2.3	tracks: 16/120 in the main set, 58/180 local-edit videos (4.5–5.2 mm, about 1 ∘ ), in every arm including the null edit
	camera dollies in	LTX-2.5	fiducial drift on 64/72 deflector videos, 12/120 track videos, 9/12 photographed track videos against 0/12 rendered twins

Table D.2 : Recurring failure forms, with the later frame of the example in Fig. G.5 where one is shown (a moved part’s first-frame position dashed). Counts are flagged videos over all videos of the experiment.

Table E.1 : Generation settings. Frames: frame rate

\times

number of frames (duration). Neg.: structural negative prompt (Sec. E.2 ). Wan 2.2 (TI2V-5B) enters only the integrity table.

Figure E.1 : Prompts. One card per prompt: where it is used, whether it names the outcome, and its text, with the colour slot highlighted.

Table F.1 : Drawn geometry on readable videos. Track: intact videos, median amplitude ratio of the drawn undulation, median on-track fraction and kinematic gain. Deflectors: intact videos, median change of the deflector angle with its 95th percentile in brackets, and median shift.

Table F.2 : Response on the track family. Top: the first three rows of Fig. G.4 , gain and phase by spatial frequency (large marks single-frequency tracks, small marks broadband components, physical value dashed, floor grey) and the speed along the straight ramp. Bottom: the complex gain, gain

|H|

and phase (degrees, positive = speed leads the drawn height), for the single-frequency levels and the three broadband components, with 95% intervals below each median.

n

: estimable videos at

f=3

,

f=4

and broadband.

Table F.3 : Dynamics on the track family: tempo, trend ratio, relaxation rate

b

, straight-ramp exponent

p

and balanced accuracy (the physical oracle of the same scenes reads 0.90–1.00), with 95% intervals below each median.

n

: estimable videos at

f=3

and on the straight level.

Table F.4 : Mechanism fits on the track family: best candidate by AIC with its parameters, largest deviation

|z|

, whether one mechanism suffices ( yes, no), and

\chi^{2}/n

(

n=30

) of every candidate at its best grid point, with the best candidate tinted. Law: the time-rescaled stated law. Cruise+s.: cruise with temporal smoothing. Target: relaxation to a slope-dependent target speed. Sat.: saturating relaxation.

Table F.5 : Contact dynamics on the deflector family. Top: exit-angle residual by contact index and first-contact restitution by layout. Bottom: restitution, exit residual and angle-response slope by model, with 95% intervals where estimated and denominators. Grey bands show the physical restitution range in the table and the measurement floors in the charts; floor values are reported in Table C.6 .

Table F.6 : Composition on the three-deflector layouts (regular and irregular pooled, prompt naming the tray). Exact order: share of trackable videos whose merged contact sequence equals the physical one, with 95% interval and counts, drawn on 0 to 100% with the physical 100% as a tick. Residual at contact

k=1,2,3

with the number of contacts. Calibrated law: median residual under the stated law and under the law fitted on the model’s single-deflector contacts, on regular and irregular layouts. Tray: share of trackable videos that end in the tray. Tinted: the one model for which the calibrated law improves the prediction. Floor of the residual 2.0–4.2 ∘ (90th percentile).

Table F.7 : Destination counterfactual on the three-deflector layouts, all named-prompt videos. Simulator: tray arrival of the stated law rolled forward from the model’s exit state after its last correct contact (videos used / wrong-route videos). Difference: model minus simulator in percentage points with 95% interval, then with the exit speed rescaled by the launch tempo and by the apparent gravity (videos used).

Table F.8 : Local edits, open-weight models (percent unless marked), pairs of the same track and seed.

dv

: speed change at the edit relative to the base video, normalized by the speed before the edit, with the unnormalized (raw) change at the dip in grey. Arrival: the normalized arrival shift of the bump over its physical value (1.07–1.10 on exact renders).

T_{\mathrm{pre}}

: change of the time before the edit, bump / dip. Path: change of the path before the edit over that under the null edit, bump / dip, with ∗ where the interval separates from the null. Null: null-edit

dv

, with † marking a flag. Floors: rerun 0 for every model except LTX-2.3 (0.15%), measurement 1.0% (

dv

) and 0.4% (

dt

).

Table F.9 : Shared-front tracks. Slope of the back-section phase on the drawn back phase (1: follows the drawing, 0: continues the front) with its resultant length

R

, drawn on

-0.5

to 1.5 with the physical 1 as a tick, and the track-family phase at

f=3

for comparison. Exploratory: locked part

|G|

and its lead, fixed part

|C|

(in brackets the physical oracle at the model’s tempo), front-section lead and its change to the back section. Front: spread of the motion over the identical front sections in mm (floor 1.6–1.7). Medians with 95% intervals.

n

: estimable videos.

Table F.10 : Energy. Top: rows 6 and 7 of Fig. G.4 , the height reached over the release height by uphill angle (one mark per video) and the share of barriers crossed by height. Bottom: ramp pairs, readable climbs. Mean height ratio over ascents with 95% interval (drawn on 0 to 4 with the physical 0.962 as a tick), ascents above the release height as dots, and per angle the median ratio over one block per climb: top stop, still climbing, stopped on the slope, turned back (top-stop ratio in the header). Physical oracle: nominal turning height with the 64-member band. Last column: flat ascents reaching the end stop. The closed models ran 60, 15 and 0 ∘ .

Table F.11 : Barriers. Per height ratio: crossings among the readable videos as dots (filled: crossed) over the median crest-speed ratio. Tinted: the barriers the stated law does not cross. Forbidden: crossings of the barriers the stated law does not cross (1.05, 1.20), with 0.95 and 0.971 counted by their nominal outcome. Missed: failures on barriers the stated law crosses. 50%: logistic crossing point. The closed models ran 0.85, 1.05 and 1.20.

Table F.12 : Chutes. Landing slope (mm per unit

\kappa

in m -1 ) with bootstrap interval and permutation

p

, mean landing offset in mm from the physical landing (720 mm), and the exit-direction slope ( ∘ per unit

\kappa

, with

p

).

n

: readable videos. Tinted: the model whose landing follows the drawn curvature.

Table F.13 : Observed readings against the two fitted mechanisms: the observed value in dark, then the predictions of the slope target and of the smoothed cruise in grey. Bump and dip:

dv

in percent.

|C|

: fixed part on shared-front tracks. Ramps: mean height ratio and the outcome on the 60 ∘ ascent (top: reaches the top stop, turn: turns back at the given ratio). Barriers: crossings (the predictions count the six heights). In brackets the family of the track-family fit.

Table F.14 : Effect of naming the destination. Top: tray arrival under named and neutral prompts. Bottom: matched prompt pairs, reporting arrival change, steering relative to the physical rollout from the model’s exit state, tempo change and the status of neutral-prompt misses. Intervals are 95% scene-bootstrap intervals.

Table F.15 : Shifted tray (neutral prompt). Capture contrast over pairs of the same base scene and seed (0: ignores the tray, 1: follows its centre), drawn on

-0.3

to 1 with the physical 0 as a tick, tray arrival in the in-tray and beyond groups (physical oracle 100% and 0%), and the median landing offset from the physical point in the two groups (mm, positive toward the deflector).

Table F.16 : Two trays (neutral prompt). Final position of trackable videos, physical tray / second tray / elsewhere, and the share of rollouts of the stated law from the model’s own exit state that end in the second tray, unscaled and scaled by the launch tempo. Physical oracle: at least 63/64 in the physical tray, 0/64 touch the second.

Table F.17 : Clip length. Top: row 9 of Fig. G.4 , the arrival at the end stop as a fraction of the clip by clip length. Bottom: tempo at 3 / 5 / 8 s (drawn on 0 to 4.5), duration exponent

\gamma

with interval and scene-seed units (drawn on

-0.1

to 1.2 with the physical 0 as a tick), the arrival at the end stop as a share of the clip over readable videos (videos), and the paired ratio of arrival times 8 s over 5 s and 3 s over 5 s (physical oracle 1, a motion fitted to the clip 1.6 and 0.6). Wan 2.2 (A14B): the tempo ratios 3 s over 5 s and 8 s over 5 s in place of

\gamma

.

Table F.18 : Track readings at 3 / 5 / 8 s: gain, phase ( ∘ ), relaxation rate

b

(s -1 , video time) with the largest length difference over its floor in brackets, and

b\alpha

(physical time, exploratory). Each line runs over the three lengths on a common scale per column (

|H|

0 to 2, phase

-20

to

+50^{\circ}

,

b

0 to 10,

b\alpha

0 to 40).

Table F.19 : Photographed first frames. Top: a photographed first frame and its rendered twin for a track and for a deflector layout. Bottom: mean difference photograph minus control over configurations, with 95% intervals below. Tracks: share of estimable videos, tempo, gain, phase ( ∘ ). Deflectors: share of trackable videos, exact order, tray arrival, largest residual ( ∘ ). ∗ , tinted: a change under the three-part rule. Control: rendered twin (open-weight models) or the nominal scene (closed models).

Table F.20 : Regeneration:

\lambda

per source version and noise level (0: stays on the source, 1: pulled onto physics), median with 95% interval and readable videos. The tint deepens with

\lambda

, from 0 to 1.2 and above (0 1.2).

Table F.21 : The defining reading of each version after regeneration, at the three levels of each model, against the source and the physical oracle.

Table F.22 : Exact sources: distance from the physical motion (mm, drawn on 0 to 220 with the physical motion at 0) and the component

\kappa

toward the model’s own image-to-video version with the same and with another seed. Last column: outputs of all versions that became the model’s own version (

\kappa\geq 0.5

), tracks and deflectors, one bar each.

Figure G.1 : Main geometry families, all models. Top: representative straight, single-frequency and broadband track scenes. Bottom: single-, regular- and irregular-deflector scenes under the named-outcome prompt. Dashed curves show physical trajectories; tagged cells mark models with no readable video of that scene.

Figure G.2 : Interventions on state, all models. Columns: the local edit (base and bump) on the one track that all nine models ran, the ramp pair with a 60 ∘ ascent, and the barriers at 0.70 and 1.20 of the release drop (the closed models ran 0.85, 1.05 and 1.20). Rows: the simulator and the nine models. The straight dashed line marks the release height and the arrow the physical turning point.

Figure G.3 : Interventions on goal and time. Left: an irregular three-deflector scene under the prompt that leaves the outcome unnamed, from the prompt pairs, for the simulator and the eight models that ran them. Right: one single-frequency track at 3 and 8 s for the four open-weight models of the clip-length arm, with the ball drawn at equal fractions of each clip, so that a motion scaled to the clip gives the same spacing at both lengths. The tagged cell (LTX-2.3) shows an altered scene.

Figure G.4 : Per-model readings of every experiment. One column per model and one row per reading, named above each row, with the physical value dashed. In the first two rows the grey band is the largest floor of the three undulating track levels at the model’s frame rate. In the last row the physical band is the track-family transit time (1.62–2.15 s) divided by the clip length, dashed at its midpoint.

Figure G.5 : Recurring integrity failures. Each pair shows the first frame and a later frame of a representative flagged video, the one nearest the median change among the flagged videos of the model shown. The later frame is the frame of largest change; for the shifted track it is shown as the amplified difference from the first frame, and for the duplicated ball it is a frame with both balls. Dashed outlines mark where the part was in the first frame. The examples show a rotating deflector, tilting tray, rotating launch ramp, shifted track, camera dolly, and ball duplication; counts and, where measured, magnitudes are reported in Table D.2 .

How Does Geometry Enter Generated Motion?

Abstract

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