A dual-representation flow-matching policy decodes each predicted motion into joint and end-effector spaces, and the residual between the two kinematically equivalent decodings provides a physically interpretable runtime signal. On multimodal tasks, however, independently sampled branches may choose different valid modes, causing false alarms. We study how to coordinate the two branches and at what cost. Across two robot environments and a non-robotic testbed, the tested mechanisms fall into four classes. An auxiliary latent shared by both branches but absent from the flow-matching construction is erased at the population optimum, a provable dead end confirmed within a prespecified 2% equivalence band. Sharing source noise can coordinate or anti-coordinate: its effect changes sign with the representation map and tracks the alignment of decoder mode basins. Consistency regularization gives intermediate coordination but reduces the valid-pair rate, while training-supported discrete partitions achieve near-ceiling coordination robustly. We further derive a chance-corrected coordination bound based only on each branch's Gini-Simpson diversity, yielding an attainable region and a label-free certificate that separates coordination from collapse when zero mismatch is ambiguous. On LIBERO-Plus, benign multimodality adds 1.57 percentage points of false alarms to the residual, which remains the strongest evaluated failure signal; the preregistered token intervention does not meet its false-alarm criterion or produce a seed-robust detection change. Code, models, and per-run configurations are available at https://github.com/kimo423/dual-head-coordination.
Figures & tables
Figure 1: Attainable region for the two robot environments ( K=2 ). The horizontal axis is modal-diversity capacity C and the vertical axis is the chance-corrected effect Δ=BL−obs , both formed per condition and aggregated with both-valid weights. The diagonal is the feasibility boundary. Each marker is one training run; fill indicates valid-pair rate. Seventeen runs have exact both-valid marginals. The remaining 50 use stored head-valid marginals and are therefore approximate; six lie at most 0.6% above the line, whereas every re-estimable condition satisfies the bound under exact accounting. The dashed path is the shared-source-noise consistency sweep.
Figure 2: Why two correct branches can disagree. The same condition is decoded into two kinematically equivalent action representations. Independent samples can select different valid routes, so the reconciled forward-kinematics residual becomes large even though both endpoints are correct. The drawing is schematic; measured decompositions appear in Section 7.1 .
numerator of Cohen’s kappa; at K=2 , 2(PLLPRR−PLRPRL)
Δ/BL
Cohen’s kappa
CK=G(q)G(p)
denominator of Gorodkin’s RK ( Gorodkin, 2004 )
ρ=Δ/CK
RK ; at K=2 , the phi coefficient/MCC ( Matthews, 1975 )
Table 1: Correspondence with chance-corrected agreement statistics (arbitrary K ).
head-valid marginals
exact both-valid marginals
excursions
Family
n
Δ
C
ρ
Δ
C
ρ
HV → BV
C zs dim-0 anchor
3
+0.0020
0.4858
+0.4%
+0.0020
0.4857
+0.4%
0/0 → 0/0
D continuous zs prior
3
−0.0006
0.4708
−0.1%
−0.0009
0.4708
−0.2%
0/0 → 0/0
E posterior (3 KL scales)
5
+0.0260
0.0256
+101.7%
+0.0223
0.0223
+100.0%
301/303 → 0/0
H episode token, uniform
3
+0.4481
0.4885
+91.7%
+0.4339
0.4766
+91.1%
3/107 → 0/0
I episode token, learned prior
3
+0.4876
0.4907
+99.4%
+0.4710
0.4810
+97.9%
68/217 → 0/0
Table 2: Exact-accounting re-estimation: head-valid versus both-valid marginals from the same contingency tables, for the 17 of 67 runs whose artifacts store both. Run-level NBV -weighted mean of per-condition quantities, then mean over seeds. Last column: per-condition excursions above C and above CK,tight , pooled over the family.
Claim
Evidence
Evidential status
∣Δ(c)∣≤CK(c)
Algebraic bound; no violation in 1275 exact-accounting robot conditions or in the testbed evaluations
Established : identity, not an empirical finding
Auxiliary latent outside the CFM construction
Two robot environments, three seeds each; TOST at δ=2% ; d^=1.16/1.66%
Supported as finite-model equivalence ; the exact zero is the population prediction under (A1)–(A2), (A3)–(A5)
Shared source noise
v2 eight-seed mean −0.5% (sd 3.4 , four signs each way); testbed factorial at ten seeds per cell: interaction range 90.1 pp, permutation p=10−4
Map dependence preregistered-and-confirmed ; seed-unstable on the robots; quote with seed and representation map
Endpoint consistency coordinates without a data-derived partition
ρ=50.3% in v2 and 63.2% in v3; valid-pair rate 0.768→0.525 in v2
Supported in the tested robot regimes , with the validity cost part of the result
Partition tokens reach the robot-side high-capacity corner
Robot-side tokens reach ρ=91.1 – 100% ; an amortized posterior reaches the same region on the balanced testbed
Supported as a robust sufficient robot-side route , not as a necessary condition in general
Benign multimodality raises residual false alarms
Residual surcharge +1.57 pp, CI [+0.70,+2.46] ; observation-side control −0.01 pp
Preregistered and supported on this benchmark and operating point
Table 3: Evidence status of the headline claims. A status applies only within the regime stated in the evidence column.
Figure 3: Mechanism summary for BimodalBypass, averaged over training seeds. The first two columns give capacity and coordination effect; the remaining columns report modal entropy, valid-pair rate, and cross-branch residual. Each column has its own scale. The near-origin posterior family illustrates why the normalized effect must be read with capacity: ρ=100% accompanies C=0.022 .
#
Family
Shared
n
obs
BL
Δ
C
ρ
Valid
Res. (mm)
Site
A
independent, β1λ0
—
3
0.4826
0.4839
+0.0014
0.4474
+0.3%
0.7729
84.79
I
B
shared source noise, β0λ0
noise
3
0.4362
0.4369
+0.0007
0.4185
+0.2%
0.7679
82.42
I ‡
C
continuous zs , dim-0 anchor
—
3
0.4921
0.4941
+0.0020
0.4857
+0.4%
0.7758
87.02
I
D
continuous zs prior, d=16
z∼N
3
0.4843
0.4834
−0.0009
0.4708
−0.2%
0.8526
85.18
I
E
continuous zs posterior (3 KL)
z∼q
5
0.0000
=C
+(0.0001 – 0.0625)
0.0001–0.0625
+100.0%
0.9292
15.78
II
F
endpoint consistency λ0.3 (+B)
loss
3
0.4235
0.4492
+0.0257
0.4363
+5.9%
0.7830
74.17
III
Table 4: Mechanism study in the BimodalBypass (v2) robot environment, using per-condition estimates; n is the number of training runs and K=2 . Sites I–IV denote the floor, near-origin boundary, interior, and high-capacity corner. Families H and I use exact both-valid marginals; the remaining rows use the stored convention stated in the supplement.
Family
n
obs
BL
Δ (per seed)
C
ρ
Valid
Resid. (mm)
A independent
3
0.4811
0.4811
0.0000 ( −0.0005/−0.0016/+0.0020 )
0.4700
0.0%
0.7382
98.5
B shared source noise
3
0.4567
0.4652
+0.0085 ( +0.0079/+0.0103/+0.0074 )
0.4555
+1.9%
0.7501
95.7
D continuous zs prior
3
0.4723
0.4748
+0.0025 ( +0.0019/+0.0019/+0.0037 )
0.4636
+0.5%
0.6474
96.9
G endpoint consistency λ=3
3
0.1709
0.4458
+0.2748 ( +0.3097/+0.2090/+0.3057 )
0.4319
+63.2%
0.4449
60.8
H episode token, uniform draw
2
0.0784
0.4708
+0.3923 ( +0.4232/+0.3614 )
0.4661
+84.1% (88.6/79.6)
0.7056
38.9
K oracle
1
0.0000
0.4994
+0.4994
0.4964
+100.0%†
0.8704
13.79
Table 5: RotatingBypass (v3) results at the coverage-matched threshold τ=0.18 . Values are per-condition estimates averaged over seeds. The ρ column averages seed-level ratios; † denotes a head-valid estimate slightly above one.
Figure 4: Run-level normalized effects ρ=Δ/C . Points are shown against zero, the measured structural-null range, and the prespecified equivalence band δ=±2% . Error bars are 95% condition-bootstrap intervals; TOST decisions use the corresponding 90% intervals. Shared source noise is shown separately because it couples the sampled flows and is not covered by Lemma 2 .
v2 family
ρˉ
sd
v3 family
ρˉ
sd
A independent (6)
+0.27%
0.27
A independent (6)
+0.00%
0.20
B shared noise (8)
−0.54%
3.41
B shared noise (8)
+0.48%
2.09
G consistency λ3 (8)
+45.3%
8.72
G consistency λ3 (6)
+41.9%
1.79
λ=1 rung (5)
+20.2%
5.18
episode token (6)
+78.5%
3.26
β=1,λ=3 control (4)
+0.58%
0.48
continuous zs prior (6)
+0.48%
0.19
episode token, learned (8) bv
+97.2%
1.36
Table 6: Preregistered seed expansion: seed-level occupancy ρ (head-valid caliber unless noted; seed count in parentheses). Between-seed sd in brackets.
Configuration
seed
obs
Valid
qL / pL
Entropy (nats)
BL
Δ
C
ρ (pool/p.-c.)
Resid.
βKL=1 , fb 0.1
0
0.0000
0.8804
0.992/0.986
0.046/0.074
0.0158
+0.0158
0.0158
104.1%/100.0%
17.61
βKL=1 , fb 0.1
1
0.0000
0.9087
0.992/0.988
0.048/0.067
0.0163
+0.0163
0.0163
102.1%/100.0%
15.95
βKL=0.01
0
0.0000
0.9620
0.990/0.991
0.057/0.051
0.0170
+0.0170
0.0170
100.2%/100.0%
14.76
βKL=0.01
1
0.0000
0.9010
0.032/0.039
0.143/0.166
0.0625
+0.0625
0.0625
100.5%/100.0%
15.56
βKL=1 , fb 0.5
0
0.0000
0.9937
1.000/1.000
0.000/0.000
0.0001
+0.0001
0.0001
100.0%/100.0%
15.04
(ref.) window token K=2
0
0.0000
0.9800
0.502/0.501
0.693/0.693
0.5000
+0.5000
0.5000
100.0%/100.0%
22.51
Table 7: Amortized-posterior runs under exact both-valid accounting, with a window-token reference. Each posterior run has obs=0 and Δ=C , so the normalized effect is 100% despite capacities close to zero. The pooled ratio is shown only to illustrate the aggregation effect.
Figure 5: Endpoint-consistency sweep. (a) Paths through the (C,Δ) plane for the β×λ grid. With shared source noise ( β=0 ), coordination increases while capacity changes little; with independent source noise ( β=1 ), the path remains near the floor. (b) Valid-pair rate for the same runs. (c) Signed normalized effects on the non-robotic testbed for K∈{2,4,8} , with three seeds per cell.
Figure 6: Two shared-variable constructions. An auxiliary random input that is irrelevant to the CFM regression target is ignored at the population optimum (top). A low-cardinality symbol fitted from training trajectories can be learned by both branches and coordinate their samples (bottom). The symbol provides a shared partition; it need not coincide with a semantic task label.
Figure 7: Effect of mode count on the balanced non-robotic testbed. (a) Normalized effects for five mechanism families across K∈{2,3,4,8} , averaged over three seeds; error bars show the seed range and the shaded line marks the prespecified 85% criterion. The unsupervised token degrades at K=4 and 8 , whereas the low-KL posterior remains above the criterion. (b) All 186 runs in relative-capacity–effect coordinates. The construction keeps capacity near its maximum; unequal target probabilities in Fig. 10 provide the complementary capacity sweep.
Figure 8: Robot-environment and balanced-testbed runs in one attainable region. (a) Raw coordinates for K=2 ; (b) the high-capacity region magnified; (c) both axes normalized by maxCK=1−1/K to compare K∈{2,3,4,8} . The balanced testbed stays near maximum capacity by construction, whereas the robot runs span the capacity axis and include the collapsed-posterior cluster. Pronounced negative effects occur only in map-dependent testbed families.
Family
K=2
K=3
K=4
K=8
amortized posterior, βKL=0.01
100.0%
—
97.1%
94.3%
amortized posterior, βKL=1 , fb 0.5
86.4%
—
75.6%
63.5%
oracle token, uniform test draw
82.4%
87.3%
84.9%
80.1%
oracle token, learned prior
83.4%
—
84.5%
80.4%
unsupervised k-means token
77.8%
78.1%
48.7%
51.2%
Table 8: Normalized coordination effect ρ across mode counts on the balanced testbed. Bold entries meet the prespecified high-capacity criterion ρ≥85% and CK≥0.80maxCK . All rows have CK≥0.992maxCK by construction.
Figure 9: Environment and representation-map checks. (a) Best single-coordinate AUROC for mode prediction in BimodalBypass and RotatingBypass; the fixed-coordinate shortcut present in v2 is substantially reduced in v3. (b) Coefficient of variation of the mode gap in the two robot environments and the non-robotic testbed. (c) Normalized effect Δ/CK under three representation maps. The floor and symbol-based families vary by less than 0.7 percentage points, whereas shared source noise and the β=0 consistency family change sign over ranges of 90.6 and 99.7 points.
Family
nullspace
orthogonal
polar
Range
independent / continuous zs prior
+0.02% / +0.10%
+0.37% / +0.30%
+0.15% / +0.17%
0.2–0.4 pp
unsupervised token / oracle token
+77.8% / +82.4%
+77.5% / +82.4%
+77.9% / +83.0%
0.4–0.7 pp
shared source noise
−22.7%
−9.2%
+67.8%
90.6 pp
β=0 consistency, λ=3
−7.3%
+10.9%
+92.4%
99.7 pp
Table 9: Occupancy ρ by representation map ( K=2 ; family pairs quoted first/second).
Figure 10: Unequal-probability testbed (153 runs, null-space map, τ=0.55 , NFE 20). (a) Relative capacity as the target prior becomes more imbalanced. Uniform token draws retain high generated diversity even when it no longer matches the target frequencies. (b) Normalized coordination and marginal distortion under uniform and learned-marginal token draws. Matching the token marginal improves both. (c) Amortized-posterior behavior over KL weight and target skew; each cell reports modal entropy and relative capacity. Strong skew with a nonzero KL term reproduces the low-capacity, zero-mismatch regime observed in the robot experiments.
Figure 11: Residual decomposition in BimodalBypass. (a) Per-condition median residuals for same-mode and cross-mode sample pairs, grouped by mechanism family. (b) Full, lateral, and label-orthogonal residual components for one uncoordinated run; the separation is concentrated in the coordinate used by the mode assignment. (c) Cross-mode contribution to the both-valid residual mean, one point per run. Stronger coordination reduces this contribution from roughly 85% to nearly zero, while the cross-mode magnitude itself remains set by the task geometry.
Quantity
v2 (10 runs)
v3 (9 runs)
same-mode floor E[D∣same]
18.0–24.0 mm
25.0–33.5 mm
cross-mode residual E[D∣cross]
151.4–159.5 mm
174.7–183.9 mm
both-valid mean E[D]
77.3–88.5 mm
87.0–105.2 mm
E[D] / same-mode floor
3.32–4.68 ×
2.96–3.96 ×
cross-mode contribution share
82.8–89.7%
81.2–87.0%
Table 11: Residual decomposition for uncoordinated families; ranges are over runs.
Object
Mechanism
obs
Δ
Valid-pair rate (uncoupled)
cross-representation (family B)
shared source noise
0.4362
+0.0039
0.768 (0.773)
ensemble b1_l0 (ten pairs)
shared x0
0.1434
+0.3632
0.9305 (0.9296)
ensemble b0_l0
shared x0
0.1859
+0.3259
0.9330 (0.9334)
ensemble b0_l3
shared x0
0.1079
+0.4003
0.8353 (0.8303)
ensemble, mixed objectives
shared x0
0.0551
+0.4130
0.8656 (0.8587)
Table 12: Effect of shared initial noise across representations and across ensemble members. Δ is reported on the pooled accounting used for this comparison; per-condition values are 0.0018 lower on average.
K
fail
succ
coord. residual
uncoord. residual
max∣z∣
RND
ACE
50
369
482
0.4878
0.5226
0.4808
0.5137
0.3080
75
369
482
0.5740
0.6212
0.5352
0.5756
0.3590
100
369
312
0.6200
0.6856
0.5285
0.5662
0.5458
125
369
99
0.6139
0.7309
0.5237
0.6066
0.5269
150
369
48
0.7667
0.9000
0.3333
0.4000
0.8667
Table 13: Fixed-horizon detection AUROC (stratified, EWMA-max aggregation, monitor seed 0) as the truncation length K varies. Failures are 250-step timeouts, so all 369 remain at every K ; successes shorter than K drop out.
Signal
Family
Stratified AUROC
uncoordinated residual, 16-sample mean
cross-representation
0.8780
uncoordinated residual
cross-representation
0.8479
coordinated residual, 16-sample mean
cross-representation
0.8401
coordinated residual
cross-representation
0.8209
random-network distillation ( Römer et al., 2025 )
observation side
0.7524
max∣z∣
observation side
0.7066
Table 14: Secondary episode-level analysis on 851 test episodes (369 timeout failures), stratified over 20 perturbation–severity cells, for the first monitor seed. Values are affected by episode length and are used only for relative ordering.
Signal
FA multimodal
FA unimodal
Gap (pp)
95% CI (pp)
coordinated residual
9.75%
8.19%
+1.57
[+0.70,+2.46]
uncoordinated residual
8.52%
6.98%
+1.54
[+0.71,+2.46]
max∣z∣
5.41%
5.42%
−0.01
[−1.17,+1.30]
random-network distillation
5.33%
6.38%
−1.05
[−2.17,+0.14]
action-chunk entropy
5.23%
5.63%
−0.39
[−1.38,+0.51]
Table 15: False-alarm rates on successful test episodes, split by the prespecified multimodality flag. Thresholds are calibrated separately for each signal at 5%; intervals use episode-level cluster bootstrap.
Token partition
Local partition
Arm
Δ
C
ρ
Δ
C
ρ
coordinated
+0.000135
0.0055
+2.5%
+0.000020
0.0281
+0.1%
independent tokens
−0.000007
0.0055
−0.1%
+0.000004
0.0282
+0.0%
shuffled pairing
−0.000003
0.0055
−0.1%
−0.000072
0.0281
−0.3%
uncoordinated network
+0.000012
0.0068
+0.2%
+0.000049
0.0309
+0.2%
prior entropy >0.30 nats
Table 16: Coordination on the LIBERO-Plus test split. Each test step is a condition; estimates use 32 samples per branch, both-valid marginals, and a three-seed mean. The last two rows retain only steps whose token-prior entropy exceeds 0.30 nats.
Appendix figures & tables8 assets
Supplementary material from the paper’s appendix.
Appendix
Round
Environment
Runs
Evaluation
Mechanism atlas
BimodalBypass (v2)
68 total
75 conditions, 500 pairs
RotatingBypass (v3)
36 conditions, 240 pairs
Threshold / NFE sensitivity
v2 and v3
(no retraining)
108 re-evaluations
Ensemble protocols
v2 only
4 families, 5 members
4 protocols, 10 pairs (main family)
Testbed, balanced
RotatingRing
186 × 50k steps
96,000 samples/run; 135 re-evaluations
Testbed, skewed prior
RotatingRing
153 (51 cells × 3 seeds)
75 conditions/run; 15.1 GPU-hours
Appendix
Table S1: Run inventory. “Runs” counts trained models; evaluation columns count what each run was scored on. The original mechanism atlas comprises 68 training runs, of which 67 carry per-condition coordinates and appear in Fig. 1 (52 v2, 15 v3). The preregistered seed expansion adds 49 robot runs and 84 testbed runs (design, seed lists, and analysis script committed before training and before unblinding, with the registration and its two revision records archived alongside the artifacts); the expansion runs are reported in Table 6 and Section 6.3 and do not enter Fig. 1 .
Family
obs
BL
Δ
C
ρ
Marg.
A independent β1λ0 (3s)
0.4826
0.4839
+0.0014
0.4474
0.3%
head
A seed0/1/2
−0.0006/+0.0029/+0.0017
0.4439/0.4495/0.4488
0.1/0.7/0.4%
head
B shared noise β0λ0 (3s)
0.4362
0.4369
+0.0007
0.4185
0.2%
head
B seed0/1/2
−0.0160/+0.0015/+0.0166
0.4565/0.3864/0.4126
−3.5 /0.4/4.0%
head
C zs dim-0 anchor (3s)
0.4921
0.4941
+0.0020
0.4857
0.4%
both
D continuous zs prior d16 (3s)
0.4843
0.4834
−0.0009
0.4708
−0.2 %
both
Appendix
Table S2: BimodalBypass (v2, 75 conditions, τ=0.11 ), per-condition accounting; Δ=BL−obs . ∗ Per-seed occupancies 100.05%/99.90%/100.02%: micro-excursions within the head-valid approximation error of Section 4.3 . ∗∗ Families H and I are quoted in the exact both-valid accounting (ratio of means), the only accounting under which they can be re-estimated from the shipped artifacts: H 91.1% (per seed 91.7/84.1/96.9) and I 97.9% (97.3/97.9/98.6), with all 75×6 conditions satisfying yi≤Ci . Under head-valid marginals the same families read 91.7% and 99.4%, and a minority of family-I conditions micro-exceed the bound (15/13/40 conditions, 68 in total, maximum +0.0463 ); family E contributes 301 excursions above C and 303 above the fixed-marginal bound across its five runs. All of them vanish under exact both-valid accounting (Section 4.5 ).
Family
obs
BL
Δ (per seed)
C
ρ
Valid
A independent (3 seeds)
0.4811
0.4811
0.0000 ( −0.0005/−0.0016/+0.0020 )
0.4700
0.3%
0.7382
B shared source noise (3 seeds)
0.4567
0.4652
+0.0085 ( +0.0079/+0.0103/+0.0074 )
0.4555
1.9%
0.7501
D continuous zs prior (3 seeds)
0.4723
0.4748
+0.0025 ( +0.0019/+0.0019/+0.0037 )
0.4636
0.5%
0.6474
G endpoint consistency λ=3 (3 seeds)
0.1709
0.4458
+0.2748 ( +0.3097/+0.2090/+0.3057 )
0.4319
63.2%
0.4449
H episode uniform token (2 seeds)
0.0784
0.4708
+0.3923 ( +0.4232/+0.3614 )
0.4661
84.1% (88.6/79.6)
0.7056
K oracle (1 seed)
0.0000
0.4994
+0.4994
0.4964
100.0% †
0.8704
Appendix
Table S3: RotatingBypass (v3, 36 conditions, τ=0.18 ), per-condition accounting; Δ=BL−obs . † 100.6% under head-valid marginals. All v3 pooled baselines are 0.4999–0.5001 (the environment is fully branch-symmetric, leaving no global preference for pooled marginals to exploit), yet pooled-accounting effects can deviate by up to 0.09; this table is per-condition throughout (Section 4.3 ). The ρ column averages per-seed occupancies (G: 65.0/56.8/67.9 → 63.2%); mean effect over mean bound gives 63.6% (ratio nonlinearity; either is usable when the aggregation convention is stated).
Run
step 1
step 25k
step 50k
prior d16 seed0
0.3791 / 0.4002
0.00280 / 0.00227
0.00146 / 0.00137
prior d16 seed1
0.4466 / 0.4674
0.00293 / 0.00256
0.00135 / 0.00127
prior d16 seed2
0.3694 / 0.4217
0.00283 / 0.00267
0.00177 / 0.00167
prior, three-seed mean
0.398 / 0.430
0.00286 / 0.00250
0.00153 / 0.00144
posterior (range, 5 runs)
0.368–0.442 / 0.382–0.525
0.0158–0.0328 / 0.0192–0.0435
0.0125–0.0215 / 0.0175–0.0269
z -dim-0 anchor (3 runs)
0
0
0
Appendix
Table S4: Sensitivity of the velocity fields to zs (relative change under a redrawn zs ; joint/position branch) at three checkpoints, from training histories. A validation-batch companion metric agrees in magnitude (prior 0.00128–0.00196; posterior 0.0122–0.0370).
λ=0
λ=0.1
λ=0.3
λ=1
λ=3
effect Δ (pooled)
shared β=0
−0.0124
−0.0062
+0.0222
+0.0799
+0.2442
partial β=0.1
−0.0117
−0.0094
+0.0088
+0.0474
+0.1519
independent β=1
+0.0024
+0.0006
+0.0038
+0.0020
+0.0126
valid-pair rate
shared β=0
0.9088
0.8728
0.8569
0.4975
0.4576
Appendix
Table S5: The full β×λ grid (v2, seed 0, 15 runs): pooled Δ=BL−obs , valid-pair rate, residual (mm).
seed
obs
95% CI
Valid
qL
pL
BL
Δ
95% CI
C
ρ
Δpc
ρpc
Hq / HT
Res.
0
0.1900
[.1853, .1949]
0.4576
0.211
0.386
0.4342
+0.2442
[ +.2382,+.2500 ]
0.3974
61.5%
+0.2404
61.7%
0.515/0.667
54.72
1
0.2782
[.2735, .2831]
0.6338
0.164
0.375
0.4160
+0.1378
[ +.1323,+.1433 ]
0.3583
38.5%
+0.1366
38.8%
0.446/0.662
58.81
2
0.2454
[.2389, .2516]
0.4849
0.562
0.611
0.4863
+0.2410
[ +.2370,+.2448 ]
0.4838
49.8%
+0.2355
49.5%
0.686/0.668
59.44
mean
0.2379
0.5254
0.4455
+0.2077
0.4132
49.9%
+0.2042
50.3%
0.549/0.666
57.66
Appendix
Table S6: Three seeds of β=0,λ=3 (v2). In the mean row the occupancy can be the mean of per-seed occupancies (pooled 49.9%, per-condition 50.0%) or mean effect over mean bound (50.3%); the text quotes the per-condition 50.3% with per-seed 61.7/38.8/49.5%.
λcons
λrot
q↔ pos
q↔ rot
pos ↔ rot
3-way
Pos. (mm)
Rot. ( ∘ )
0
0
+0.0501 [ +0.0316 , +0.0633 ]
+0.0227 [ +0.0064 , +0.0371 ]
+0.0291 [ +0.0070 , +0.0476 ]
0.307
93.90
13.65
0
1
+0.0403 [ +0.0241 , +0.0522 ]
+0.0223 [ +0.0084 , +0.0355 ]
+0.0356 [ +0.0170 , +0.0523 ]
0.302
94.00
13.46
3
0
+0.3353 [ +0.3210 , +0.3465 ]
+0.0018 [ −0.0131 , +0.0165 ]
+0.0279 [ +0.0087 , +0.0461 ]
0.430
60.97
13.05
3
1
+0.2671 [ +0.2499 , +0.2843 ]
+0.0227 [ +0.0105 , +0.0337 ]
+0.0166 [ +0.0009 , +0.0316 ]
0.408
63.40
12.63
Appendix
Table S7: λcons×λrot ablation (v3, seed 0, τ=0.18 ; the orientation branch is mode-assigned by nearest mean-yaw match; Δ=BL−obs . Random three-way agreement 0.25; teacher-forced training residuals: position 10.7–17.6 mm, orientation 1.45–1.88 ∘ ).
τ
v2 Δ
v2 valid
v3 Δ
v3 valid
0.05
+0.1913
0.729
+0.2150
0.999
0.08
+0.2141
0.609
+0.2150
0.999
0.11
+0.2442
0.458
+0.2161
0.994
0.14
+0.2995
0.255
+0.2396
0.880
0.17
+0.4427
0.050
+0.3006
0.585
0.20
—
—
+0.3936
0.154
Appendix
Table S8: λ=3 family under threshold variation (pooled Δ ; valid-pair rate).
Multimodal world models are often evaluated by whether different sensors produce similar representations. However, similar representations do not necessarily imply that the models make the same physical predictions, or that those representations can be reused when actions are combined in a new order. We study both questions through the physical responses predicted by a model. We first use the Cluster Haptic dataset to ask whether audio and acceleration can independently recover the behavior of the same surface from different observations. Predictions from the two sensors are substantially closer for the same surface than for different surfaces, with a 4.5× gap on average, while both also outperform an average-surface prediction. We then show that this agreement alone does not determine how familiar actions should compose. In a controlled elastoplastic system, shared step dynamics fit observed programs less accurately than a whole-program predictor but generalize better to unseen action orders, with the ranking reversing on both held-out transitions across three independent initializations. Fusing free-decay and hysteresis observations further improves prediction, with diagonal Gaussian beliefs yielding the lowest errors. Together, these results distinguish cross-sensor consistency, multimodal fusion, and generalization to new action orders as separate questions in evaluating multimodal physical representations.
Kaizhen Tan, Xin Xu, Siru Tao +4
New York University, New York, NY, USA · Carnegie Mellon University, Pittsburgh, PA, USA · Columbia University, New York, NY, USA
In robotic manipulation, the same observation often admits multiple feasible actions. Generative policies such as diffusion and flow matching naturally capture such multimodal behaviors, yet which mode is executed is left to random sampling and cannot be actively controlled. We propose Mode-Oriented Source Shaping (MOSS), which turns mode selection into an explicit choice of where generation starts. We first reveal, both empirically and theoretically, that deterministic flow transport induces mode-dependent structure in the source space. Under ideal deterministic ODE transport, distinct behavioral modes have distinct source-space preimages, while practical approximation errors can blur their boundaries due to imbalanced data, limited model capacity, and few inference steps. Motivated by this source-space structure, MOSS constructs semantically aligned mode-specific Gaussian sources that encode mode identity in the initial noise. A modality-strength estimator predicts the local degree of multimodality, which adaptively controls the geometry of the mode-specific sources. It contracts them toward a shared Gaussian where behaviors are similar and separates them near behavioral branching points. During deployment, the desired behavior is selected at inference simply by sampling from the corresponding component. Across 6 simulated and 3 real-world manipulation tasks, MOSS steers to the target mode up to 100% of the time with high task success, recovers minority modes at over 90% guidance rate under a 10:90 imbalance dataset, and surpasses mode-conditioned flow policy by 43.3 points of guidance rate on real robots.
Zirun Ma, Jindou Jia, Tuo An +3
MARS Lab, Nanyang Technological University · Tsinghua University
Generative modeling is widely used for producing diverse objects from complex, multimodal distributions. However, its expressivity does not, in general, come with formal guarantees that the generated objects satisfy hard constraints or requirements. In multi-agent generation, this problem becomes more challenging because a hard requirement can depend on multiple agents, while each agent may need to determine its own guidance input without relying on the simultaneously computed guidance inputs of other agents. To this end, we introduce DeGG-Flow, a general framework for multi-agent flow matching with decoupled generative guidance. By representing the generative process as a control-affine dynamical system, we develop guidance conditions for two classes of coupled requirements: shared requirements whose satisfaction depends on multiple agents together, and private requirements associated with each individual agent dependent on its neighbors. For both classes, we establish feasibility conditions and finite-horizon convergence guarantees. We further derive a Wasserstein bound that characterizes the distributional deviation induced by the guidance. We demonstrate DeGG-Flow on multi-robot collaboration for crossing a spatial gap by reconfiguring the environment, and on multi-object scene generation with affordance requirements. Across both applications, DeGG-Flow directly generates objects that satisfy all corresponding hard requirements, including at team sizes unseen during training.
Ruoyu Lin, Magnus Egerstedt, Fabio Pasqualetti
Department of Electrical Engineering and Computer Science, University of California, Irvine, Irvine, CA 92697 USA · University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 USA