Shaping Persistent Representations from Independent Interactions

Ji Dai; Quan Fang; Junyu Gao; Rongfeng Guo; Haoyan Rong; YipingHuang; Yongxi Li

cs.LGSep 28, 2026

Shaping Persistent Representations from Independent Interactions

Authors: Ji Dai, Quan Fang, Junyu Gao, Rongfeng Guo, Haoyan Rong, YipingHuang, Yongxi Li

Organizations: Beijing University of Posts and Telecommunications · Institute of Automation, Chinese Academy of Sciences · Shenzhen University · Tsinghua University

Abstract

World models learn environment dynamics from interaction experience. These dynamics depend on the current state and actions, as well as on properties that persist across interactions. Yet standard predictive training can reduce error using local evidence alone, without organizing persistent information into reusable context. We introduce SPRII, a training principle that uses relations between interactions as weak supervision for persistent context while retaining the learner's native objective. For example, different trajectories of the same system share persistent properties even when their states and actions differ. SPRII uses such relations to guide context learning without numerical property labels. Two composable components encourage contexts from related interactions to agree (Align) and use one interaction's context to predict another's future (Cross). Our analysis distinguishes three linked questions: what persistent information is accessible in the learned context (Formation), how that context influences a fixed predictor (Use), and whether it reduces task error (Value). Success at one stage does not guarantee success at the next. Controlled experiments show that more reliable relations improve representation organization, but adding a shared-property constraint can reduce access to a property that remains shared. Context substitutions change predictions at fixed model weights, while the benefit from history depends on prediction horizon and readout. Evaluations span thirteen settings, including controlled physical systems, public dynamics tasks, robotic and tactile data, and partner interaction, across multiple learner families. Relative to the corresponding baselines, SPRII yields average gains of over 10% in downstream task performance and over 15% in persistent-property readout. The project page is available at https://persistent-learning-review.netlify.app/interactive.html.

Figures & tables

Figure 1: From related interactions to useful persistent information. All panels are schematic. (a) Interactions differ in state

s_{t}

and action

a_{t}

while sharing mass, drag and stiffness,

\theta=(m,\gamma,k)

. (b) Formation organizes persistent information. (c) Use tests its effect on a fixed reader’s predictions. (d) Value tests whether it lowers task error. The linked questions distinguish information formation, predictive use and task benefit; success at one stage does not guarantee the next.

Figure 2: Relation-supervised learning of persistent context. Thick arrows carry recipient inputs (a) into both prediction branches (b). (a) Align regularizes contexts paired by

\mathcal{R}_{G}

. (b) Both calls share predictor

q

, current input

C_{B}

and conditioning

u_{B}

; Cross replaces only

z_{p}^{B}

with related donor context

z_{p}^{A}

y_{B}

is the loss target. Both components augment the native objective. Trajectories and tokens are schematic; Figure 8 shows the JEPA implementation.

A. SpringWorld: single-state queries, 256 sealed systems
Source recipe	MSE $\downarrow$	SPRII reduction	95% CI
Native JEPA	$0.138\,\text{$ \pm,0.002 $}$	22.3%	[13.2, 30.7]%
TDS	$0.130\,\text{$ \pm,0.013 $}$	17.5%	[8.4, 25.9]%
SPRII (Align + Cross)	$\mathbf{0.107}\,\text{$ \pm,0.004 $}$	Reference

Table 1: Primary predictive performance on SpringWorld and CoPhy. Lower is better; bold marks the lowest comparable mean. 1

A. Burgers: native-trajectory / zero-step MSE ( $\times 10^{-3}$ )
Host learner	Source recipe	ID	Viscous OOD	Inviscid OOD
NOD (arXiv'26)	Released configuration	$9.096\,\text{$ \pm,0.544 $}$	$8.012\,\text{$ \pm,0.815 $}$	$11.857\,\text{$ \pm,0.628 $}$
	+ SPRII	$\mathbf{8.999}\,\text{$ \pm,0.798 $}$	$\mathbf{7.487}\,\text{$ \pm,0.183 $}$	$\mathbf{11.510}\,\text{$ \pm,0.360 $}$
Amortized CoDA (ICML'22)	Plain	$3.483\,\text{$ \pm,0.567 $}$	$6.390\,\text{$ \pm,1.436 $}$	$5.883\,\text{$ \pm,1.700 $}$
	+ SPRII (Cross)	$\mathbf{3.250}\,\text{$ \pm,0.729 $}$	$\mathbf{6.060}\,\text{$ \pm,1.185 $}$	$\mathbf{5.501}\,\text{$ \pm,1.496 $}$

Table 2: Predictive comparisons on Burgers and D-Clean. Lower is better; bold marks the lowest mean within each comparable host or reader group. 2

Figure 3: Relations shape the strength and content of persistent representations. (a) Pair reliability strengthens separation and drag geometry; whiskers: source SD. (b) Partial geometry. (c) Linear/MLP probe

R^{2}

G_{1}=m

G_{2}=(m,\gamma)

G_{3}=(m,\gamma,k)

. Means cover three sources. Geometry uses 400 confirmation systems; probes use 400 validation systems (Appendix E.1 ).

Figure 4: Donor physics structures fixed predictive responses. (a)–(c) Donor–target landscapes of h16 standardized-increment MSE, averaged over 64 development systems and three Align + Cross sources; color scales differ. Only donor context changes the fixed reader’s prediction. (d) Mean off-diagonal minus diagonal error; positive favors matching. Points: sources; diamonds: means; intervals: pointwise 95% paired-system CIs, conditional on fitted models (Appendix F.1 ).

Figure 5: Persistent context has prediction- and readout-dependent value. (a) SpringWorld gain over Null readers as horizon increases; whiskers: pointwise 95% system-bootstrap intervals over 178 systems. (b) Scaling context in fixed

M_{2}

changes MSE relative to recipient-only

R_{0}

;

s=0

gives

R_{0}

. (c1) Query-only

M_{1}

versus context-conditioned

M_{2}

: source means over three readers. (c2) MSE change from Decode to Persistent (%); negative favors direct context. In (c2), endpoints are D-Clean h32 and SpringWorld cold-start h16 (Appendix G ).

Appendix figures & tables91 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Selected task and property gains. Left: downstream task improvement. Right: improved readout of the labeled property. Both are relative to the corresponding baseline. Asterisks mark single-source examples; Figure 7 gives comparison details.

Finding	Broader implication	Evidence
• More reliable relations strengthen the organization of shared information.	How experiences are linked may matter alongside how much data is collected.	App. E.1
• Sharing more properties can make a still-shared property harder to read out.	A useful relation may be one that emphasizes the information a task needs.	App. E.1
• In SpringWorld, alignment drives much of the organization; cross-prediction adds further task gains.	Learning what should stay consistent and learning how to put it to work are distinct goals that can reinforce each other.	App. E.2
• Changing the supplied history redirects the same predictor along the changed physical properties.	A representation’s role becomes clearer when we examine how decisions respond to it.	App. F.1 ; F.2
• In SpringWorld, history brings larger gains at longer prediction horizons under the same current-evidence budget.	Past experience may be most useful when it resolves uncertainty left by the present situation.	App. G.1
• Direct context works better in D-Clean; decoded physical parameters work better in SpringWorld.	A more interpretable representation need not be the most useful interface for a task.	App. G.2

Appendix

Table 3: Findings and possible design implications. The evidence column links each observation to its supporting experiments.

Setting	Relation	Experience $\to$ prediction	Evaluated instance	Primary evidence
Controlled environments
SpringWorld	Shared: Mass $m$ , drag $\gamma$ , stiffness $k$ . Varied: Interaction history.	Input: Independent history; current query/actions. Target: Future state increments.	Learner: JEPA. Instance: Align + Cross. Controls: Native; TDS. Components: Structure; Align; Cross.	Primary: Sealed new-system prediction . Further: Geometry ; Use ; Value .
PokeWorld	Shared: Selected factors: $m$ ; $(m,\gamma)$ ; $(m,\gamma,k)$ . Varied: Correct-pair fraction $\alpha$ , separately.	Input: Independent image/action history; current query. Target: Future observation embedding.	Learner: JEPA. Instances: Structure; Align; Cross; Align + Cross. Controls: Relation and objective composition.	Primary: Formation: relation reliability and shared factors . Further: Probes ; Use ; task readouts .
D-Clean	Shared: Drag $\gamma$ . Varied: Initial state; force sequence.	Input: Independent state/action history; recipient state. Target: Future prediction.	Learner: JEPA. Source: Align + Cross. Readouts: Null; Persistent; Decode; Oracle. Comparisons: TDS; NOD; FCRL; CaDM; DALI.	Primary: Formation ; Value . Further: Closest methods .
Public benchmarks and real data
CoPhy Collision	Shared: Matched objects; scene-specific rule. Varied: Support interaction.	Input: Three support histories; three query frames. Target: Future trajectory.	Learners: CPC; RSSM; JEPA. Instance: Cross. Controls: Native; Structure; Random. Extensions: Other learners in H.1 .	Primary: Held-out prediction . Further: Learner matrix .

Appendix

Table 4: Evaluation settings and their principal tests. Images illustrate the settings, not model predictions.

Figure 7: Original-value task and property comparisons. Each setting pairs task performance (left) with property readout (right). Blue: reference; orange: SPRII. Arrows mark the preferred direction.

Inputs: recipient sampler $P$ , donor kernel $Q_{G}$ , encoders and task readout $q$ .
1	Sample recipients and related donor histories:
	$B\sim P,\qquad H_{A}\sim Q_{G}(A\mid B)$ .
2	Encode the histories and current recipient input:
	$z_{\mathrm{p}}^{A}=g_{\mathrm{p}}(H_{A}),\qquad z_{\mathrm{p}}^{B}=g_{\mathrm{p}}(H_{B}),\qquad z_{\mathrm{s}}^{B}=g_{\mathrm{s}}(C_{B})$ .
3	Compute the native objective $\mathcal{L}_{\rm base}$ on ordinary recipient inputs.
4	Initialize $\mathcal{L}_{\mathrm{align}}\leftarrow 0$ , $\mathcal{L}_{\mathrm{cross}}\leftarrow 0$ .

Appendix

Algorithm 1 One SPRII training update.

Figure 8: Controlled-JEPA implementation of SPRII. (a) Related observation/action histories. (b) A shared observation encoder feeds separate current and persistent branches; Align compares the persistent codes. (c) Self and Cross share the predictor and recipient target. Cross replaces only the persistent input with the donor’s code. Self-prediction and SIGReg together form the native objective in Figure 2 . Tokens and the causal-mask crop are schematic.

Instance	Native predictive route and measured representation	Relation operations
• Controlled JEPA	24-step history $P_{64}$ , current $S_{64}$ ; future online embedding MSE + SIGReg. D-Clean state MLP; Poke visual CNN.	Structure, Align, Cross, Align + Cross.
• Spring JEPA	96-frame visual/action history; embedding MSE + SIGReg; frozen source and separately fitted physical readers.	Align + Cross, with component controls.
• CoPhy JEPA/CPC	Frozen 784D perception features; latent prediction or future contrast; native dynamics state retained.	Cross in the main matrix.
• CoPhy RSSM	Reconstruction, prior prediction and KL; $U=[P_{64},h_{128},z_{32}]$ , versus 160D Native $U$ .	Cross; $U$ , $S$ , $M$ measured separately.
• CoPhyNet	Coordinate MSE and stability classification; original 32D code split as $P_{16},T_{16}$ ; reader receives full $U$ .	Align + Cross on $P$ .
• Baxter	40-step tactile history, $P_{64},S_{64}$ ; future embedding MSE + SIGReg.	Align + Cross; hardness/shape relations.

Appendix

Table 5: Predictive routes and relation operations across learners.

U

denotes complete source state,

S

independent support and

M

trained reader memory; each is probed separately.

Figure 9: External field environments. (a) A released Burgers training trajectory at $\nu=0.01$ . (b) A separately simulated FHN trajectory at $k=0.03,\beta=0.20$ . FHN panels share a signed field scale; Burgers uses a separate scale. These are environment illustrations, not prediction results.

Term	Meaning
Persistent context	History-derived representation supplied to a predictor; source weights are frozen for the stated diagnostic evaluations.
Probe	Supervised diagnostic fitted to a frozen code; measures accessibility.
Task reader	Predictor fitted on fixed source features; measures realized task value.
Fixed-predictor intervention	Donor replacement with all source and consuming weights fixed.
Native	History-conditioned learner with its original context interface and objective.
Structure	Separate persistent/current interface, without relation objectives.

Appendix

Table 6: Terminology used in the main text and appendix.

Analysis	Population	Source, reader and measurement
SpringWorld
Sealed comparison	256 continuous primary; 36 factorial secondary	$3$ sources $\times$ $3$ readers; matched inputs. The 292-system pool is auxiliary.

Original reader	100-system cold start; 178-system mixture	Separate grids: source 0/reader 0 reference; three-source component mean with reader 0; original $3\times 3$ comparison.

Common adapter	100-system bottleneck; 178-system horizon/probes	FCRL uses each source’s own query encoder; the four bottleneck arms share one source’s query encoder.

Appendix

Table 7: Evaluation populations, fitted-model coverage and consuming routes.

Experiment	Operation	Seed
Baxter	Grasp-identifier split permutation	20260822
RH20T additional 25-task bank	Paired task bootstrap	20260923

Appendix

Table 8: Fixed randomization settings used in the appendix.

Source	Primary: 256 systems	Secondary: 36	Pooled: 292
A. Absolute prediction error
Native	0.13820 $[0.11850,0.15920]$	0.21370	0.14750
TDS	0.13030 $[0.11180,0.15020]$	0.19670	0.13840
Align + Cross	0.10740 $[0.09036,0.12670]$	0.12730	0.10990

Appendix

Table 9: Sealed SpringWorld prediction and paired contrasts. Results average three sources and three readers per source. Brackets show 95% physical-system intervals conditional on these fitted models. Positive gaps favor Align + Cross; the system-weighted pooled result is auxiliary.

Figure 10: Cross lowers RSSM error on most held-out episodes. (a) Error distributions on 1,994 episodes. (b)–(d) Paired comparisons with each control. Episode MSEs are averaged over three source/reader runs. Points below the diagonal favor Cross; fractions report their share. Axes are logarithmic.

Family	Native	Structure	Random	SPRII	vs. Structure	vs. Random
	Physical trajectory MSE				Reduction
CPC	$0.248\,\text{$ \pm,0.008 $}$	$0.223\,\text{$ \pm,0.006 $}$	$0.221\,\text{$ \pm,0.006 $}$	$\mathbf{0.189}\,\text{$ \pm,0.007 $}$	15.4%	14.7%
RSSM	$0.268\,\text{$ \pm,0.007 $}$	$0.228\,\text{$ \pm,0.005 $}$	$0.222\,\text{$ \pm,0.007 $}$	$\mathbf{0.202}\,\text{$ \pm,0.007 $}$	11.4%	9.0%
JEPA	$0.250\,\text{$ \pm,0.004 $}$	$0.215\,\text{$ \pm,0.007 $}$	$0.265\,\text{$ \pm,0.003 $}$	$\mathbf{0.184}\,\text{$ \pm,0.005 $}$	14.2%	30.4%

Appendix

Table 10: Held-out Collision prediction. Physical trajectory MSE on 1,994 eligible episodes, mean

\pm

sample SD across three joint source/reader seeds. Training budgets match within each learner family.

Method	MSE	Source / reader epochs
Native	$1.292\,\text{$ \pm,0.037 $}$	100 / 100
Structure	$1.280\,\text{$ \pm,0.031 $}$	100 / 100
Random	$1.327\,\text{$ \pm,0.011 $}$	100 / 100
SPRII (Cross)	$1.154\,\text{$ \pm,0.025 $}$	100 / 100

Appendix

Table 11: RSSM on 1,000 held-out Balls4 recipients. MSE: mean

\pm

sample SD over three seeds.

Source	Updates	h1	h4	h16	h32
SPRII	20,000	$0.0314\,\text{$ \pm,0.002915 $}$	$0.1166\,\text{$ \pm,0.005082 $}$	$0.5232\,\text{$ \pm,0.01173 $}$	$1.507\,\text{$ \pm,0.1355 $}$
CaDM	20,000	$0.03324\,\text{$ \pm,0.001838 $}$	$0.1474\,\text{$ \pm,0.008051 $}$	$0.7711\,\text{$ \pm,0.008115 $}$	$2.273\,\text{$ \pm,0.06284 $}$
FCRL	20,000	$0.1228\,\text{$ \pm,0.01122 $}$	$0.762\,\text{$ \pm,0.08086 $}$	$4.833\,\text{$ \pm,0.492 $}$	$13.34\,\text{$ \pm,1.014 $}$
NOD	20,000	$0.3894\,\text{$ \pm,0.6261 $}$	$3.814\,\text{$ \pm,6.412 $}$	$23.79\,\text{$ \pm,40.18 $}$	$61.88\,\text{$ \pm,104.1 $}$
DALI	20,000	$0.5579\,\text{$ \pm,0.02027 $}$	$5.307\,\text{$ \pm,0.2763 $}$	$35.49\,\text{$ \pm,1.735 $}$	$89.3\,\text{$ \pm,3.886 $}$
DALI	80,000	$0.3314\,\text{$ \pm,0.035 $}$	$2.962\,\text{$ \pm,0.2441 $}$	$20.54\,\text{$ \pm,1.948 $}$	$53.9\,\text{$ \pm,5.067 $}$

Appendix

Table 12: D-Clean prediction with a common reader. Raw-state MSE in units of

10^{-4}

, mean

\pm

source SD after averaging three readers per source (

3

sources). Source budgets vary as shown. The 100-system report population was previously evaluated; it is not a new sealed test.

Reader	Source	Fresh Null	Matched	Fixed Wrong
Original	Align + Cross	0.1775	0.1121	0.2439
Original	TDS	0.1716	0.1596	0.1787
Original	Compact TDS ( $d=0$ )	0.1709	0.1709	0.1709

Appendix

Table 13: Prediction with adapted baselines and context controls. A/B report Spring standardized MSE; C reports D-Clean raw-state MSE, each averaged equally over three sources and three readers. Fixed Wrong uses the Matched reader. D uses native TDS decoders rather than C’s common reader; its direct and rollout errors are separate diagnostics.

Primary comparison
Track	ID	Viscous OOD	Inviscid OOD
Released-default NOD	$9.096\,\text{$ \pm,0.544 $}$	$8.012\,\text{$ \pm,0.815 $}$	$11.86\,\text{$ \pm,0.628 $}$
SPRII fixed endpoint	$\mathbf{8.999}\,\text{$ \pm,0.798 $}$	$\mathbf{7.487}\,\text{$ \pm,0.183 $}$	$\mathbf{11.51}\,\text{$ \pm,0.360 $}$
Matched additional-update controls
Continued NOD	$9.379\,\text{$ \pm,0.340 $}$	$7.408\,\text{$ \pm,0.130 $}$	$11.49\,\text{$ \pm,0.215 $}$
Random continuation	$9.538\ (\text{--})$	$8.900\ (\text{--})$	$11.50\ (\text{--})$

Appendix

Table 14: Burgers prediction and continuation controls. Native trajectory MSE in units of

10^{-3}

, mean

\pm

sample SD across three sources. SPRII includes prediction-only continuation; controls match its added updates. Released-default NOD has no added updates. Dashes denote unreported source SD.

	ID B/W	ID $R^{2}$	$H=50$ MSE ( $\times 10^{-3}$ )
Method			Matched	Wrong $-$ matched
NOD	$22.3\,\text{$ \pm,19.09 $}$	$0.651\,\text{$ \pm,0.458 $}$	$12.89\,\text{$ \pm,0.992 $}$	$0.014\,\text{$ \pm,0.391 $}$
SPRII	$\mathbf{31.64}\,\text{$ \pm,4.506 $}$	$\mathbf{0.762}\,\text{$ \pm,0.141 $}$	$\mathbf{12.19}\,\text{$ \pm,1.944 $}$	$\mathbf{-0.054}\,\text{$ \pm,0.597 $}$

Appendix

Table 15: Burgers organization, viscosity accessibility and donor use. Entries are mean

\pm

SD across three sources. B/W is the latent between/within squared-distance ratio. A uses 45 ID recipients and a direct MSE donor assay; B uses training geometry and a separate 11-system relative-

L_{2}

assay. Positive penalties indicate harm from wrong donors; 95% intervals resample paired sources and systems.

Method	ID	Viscous OOD	Inviscid OOD	Code steps
GEPS	$5.60\,\text{$ \pm,7.05 $}$	$12.37\,\text{$ \pm,14.21 $}$	$13.84\,\text{$ \pm,17.41 $}$	50
CoDA	$3.38\,\text{$ \pm,0.57 $}$	$6.48\,\text{$ \pm,1.42 $}$	$5.89\,\text{$ \pm,1.55 $}$	50

Appendix

Table 16: Burgers prediction with test-time adaptation. Native full-trajectory MSE in units of

10^{-3}

, mean

\pm

sample SD across three sources. Both methods use 50 support-code updates per query; zero-step results are in Table 14 .

Arm	Objective addition	Role
Plain amortized CoDA	Self prediction and common regularization	History-encoder baseline.
Matched Cross	A second prediction using the related donor	Relation-supervised arm.
Random Cross	The same second prediction using a wrong-system donor	Pairing control with matched donor marginals.
Duplicate Self	A second self-prediction forward/backward pass	Matched computation and reconstruction-loss scale.

Appendix

Table 17: Objectives on the frozen CoDA decoder.

Distribution	Arm	Code-0	Code-50
ID	Plain	$3.4834\,\text{$ \pm,0.5669 $}$	$\mathbf{3.5123}\,\text{$ \pm,0.5353 $}$
ID	Matched Cross	$\mathbf{3.2500}\,\text{$ \pm,0.7286 $}$	$3.5566\,\text{$ \pm,0.4909 $}$
ID	Random Cross	$3.3076\,\text{$ \pm,0.7055 $}$	$3.5164\,\text{$ \pm,0.4933 $}$
ID	Duplicate Self	$3.4203\,\text{$ \pm,0.7573 $}$	$3.5899\,\text{$ \pm,0.4809 $}$
Viscous OOD	Plain	$6.3900\,\text{$ \pm,1.4357 $}$	$7.1890\,\text{$ \pm,1.6605 $}$
Viscous OOD	Matched Cross	$\mathbf{6.0598}\,\text{$ \pm,1.1847 $}$	$7.0845\,\text{$ \pm,1.4797 $}$

Appendix

Table 18: Amortized prediction with a frozen CoDA decoder. Native full-trajectory MSE in units of

10^{-3}

, mean

\pm

sample SD across three sources. Code-0 is the primary endpoint.

Distribution	Comparator	Seed 1	Seed 2	Seed 3	Mean [95% CI]	$n_{+}$
ID	Plain	0.1270	0.4228	0.1502	$0.2333\,[-0.1752,0.6418]$	3
ID	Random Cross	$-0.0113$	0.0710	0.1129	$0.0576\,[-0.0994,0.2145]$	2
ID	Duplicate Self	0.3036	0.1917	0.0155	$0.1703\,[-0.1904,0.5310]$	3
Viscous OOD	Plain	0.1989	0.7125	0.0793	$0.3302\,[-0.5055,1.1659]$	3
Viscous OOD	Random Cross	$-0.0904$	0.1676	$-0.0071$	$0.0234\,[-0.3038,0.3506]$	1
Viscous OOD	Duplicate Self	0.4304	0.8332	0.0499	$0.4378\,[-0.5351,1.4108]$	3

Appendix

Table 19: Paired CoDA contrasts at code-0. Differences are comparator minus Matched Cross in units of

10^{-3}

; positive favors Matched Cross. Intervals are paired-source 95%

t

intervals (

n=3

);

n_{+}

counts positive source differences.

Distribution	Comparator	Seed 1	Seed 2	Seed 3	Mean [95% CI]	$n_{+}$
ID	Plain	0.0046	$-0.0930$	$-0.0446$	$-0.0443\,[-0.1655,0.0768]$	1
ID	Random Cross	$-0.0361$	$-0.0165$	$-0.0681$	$-0.0402\,[-0.1049,0.0244]$	0
ID	Duplicate Self	0.0232	0.0760	0.0006	$0.0333\,[-0.0629,0.1294]$	3
Viscous OOD	Plain	0.1935	0.2277	$-0.1076$	$0.1045\,[-0.3537,0.5628]$	2
Viscous OOD	Random Cross	$-0.0976$	0.1018	$-0.1646$	$-0.0534\,[-0.3977,0.2908]$	1
Viscous OOD	Duplicate Self	0.3286	0.5325	0.0882	$0.3165\,[-0.2360,0.8689]$	3

Appendix

Table 20: Paired CoDA contrasts at code-50. Differences are comparator minus Matched Cross in units of

10^{-3}

; positive favors Matched Cross. Intervals are paired-source 95%

t

intervals (

n=3

);

n_{+}

counts positive source differences.

Arm	Code-0 viscosity $R^{2}$	Code-50 viscosity $R^{2}$
Plain	$0.1325\,\text{$ \pm,0.1388 $}$	$0.2765\,\text{$ \pm,0.2184 $}$
Matched Cross	$\mathbf{0.1752}\,\text{$ \pm,0.1545 $}$	$\mathbf{0.3010}\,\text{$ \pm,0.1946 $}$
Random Cross	$0.1291\,\text{$ \pm,0.1324 $}$	$0.2845\,\text{$ \pm,0.1945 $}$
Duplicate Self	$0.1527\,\text{$ \pm,0.1604 $}$	$0.2878\,\text{$ \pm,0.2334 $}$

Appendix

Table 21: CoDA viscosity accessibility and decoder invariance. Probe

R^{2}

is mean

\pm

sample SD across three sources on 45 ID histories. Decoder change is the maximum absolute change during encoder fitting, over all three decoders; all remain unchanged.

Quantity	Plain	Matched Cross	Random Cross	Duplicate Self
Trainable encoder parameters	89,282	89,282	89,282	89,282
Encoder fitting time	1267.92	2489.37	2517.80	2519.17
Code-0 encoder + forecast	0.03416	0.03500	0.03510	0.03353
Code-50 adaptation time	6.84052	6.74525	6.57095	6.47388
Code-50 forecasting time	0.03490	0.03604	0.03407	0.03370
Shared decoder fitting time	5372.08

Appendix

Table 22: CoDA fitting and evaluation costs. Times are seconds, averaged over three sources: per fit for training and per query for evaluation. Code-0 includes encoder inference and forecasting; Code-50 rows exclude encoder inference. Evaluation times are amortized execution costs, excluding data transfer, decoder construction and scoring. Shared decoder and teacher costs are reported separately.

Quantity	Native	Align + Cross
Source optimizer updates	10,000	10,000
Additional continuation updates	0	0
Reader updates per fitted reader	10,000	10,000
Trainable source parameters	4,509,120	4,525,632
Compute-only time per source update (s)	0.1756	0.1948
Original full-fit wall time (s)	34,270.26	34,324.20

Appendix

Table 23: SpringWorld training cost per fit. Update times use a common RTX 6000D benchmark. Full-fit times average three original runs under different hardware and runtime conditions and are not directly comparable to the update benchmark.

Comparison	Source	Reader	Configuration and selection
Native / Structure / Align / Cross / joint	10,000	10,000	Fixed component weights; final source step. Component table uses reader 0.
Sealed Native / TDS / joint	10,000	10,000	Three sources $\times$ three readers; sources and readers frozen before the primary test.
Full / compact TDS	10,000	10,000	$d\in\{0,\ldots,6\}$ for compact TDS; separate selection-development bank.
Temporal FCRL-style adaptation	10,000	10,000	Fixed temperature 0.07; three sources $\times$ three common-adapter readers.

Appendix

Table 24: SpringWorld budgets and configuration selection. Updates are per fit.

$\lambda_{\mathrm{p}}$	$\lambda_{\mathrm{x}}$	Dev. MSE	Mass $R^{2}$	Drag $R^{2}$	Stiffness $R^{2}$
0.25	0.1	$0.1203\,\text{$ \pm,0.0025 $}$	$0.2837\,\text{$ \pm,0.0494 $}$	$0.1441\,\text{$ \pm,0.1210 $}$	$-0.0204\,\text{$ \pm,0.0218 $}$
1	0.1	$0.1100\,\text{$ \pm,0.0081 $}$	$0.3630\,\text{$ \pm,0.0408 $}$	$0.1598\,\text{$ \pm,0.0220 $}$	$0.1430\,\text{$ \pm,0.0284 $}$
4	0.1	$0.1609\,\text{$ \pm,0.0360 $}$	$0.3498\,\text{$ \pm,0.0989 $}$	$0.2694\,\text{$ \pm,0.1131 $}$	$0.3166\,\text{$ \pm,0.2892 $}$
1	0.025	$0.1049\,\text{$ \pm,0.0112 $}$	$0.4162\,\text{$ \pm,0.0846 $}$	$0.1645\,\text{$ \pm,0.0447 $}$	$0.3578\,\text{$ \pm,0.2626 $}$
1	0.4	$0.1274\,\text{$ \pm,0.0027 $}$	$0.3751\,\text{$ \pm,0.0322 $}$	$0.1331\,\text{$ \pm,0.0277 $}$	$0.2403\,\text{$ \pm,0.1694 $}$

Appendix

Table 25: Sensitivity to Align and Cross weights. Mean

\pm

SD over three sources; MSE first averages three readers per source. Probes predict

\log m

\log\gamma

and

\log k

$(\lambda_{\rm p},\lambda_{\rm x})$	Valid $n$	Mean	Q1	Median	Q3	Negative (%)
$(0.25,0.1)$	300	0.0263	$-0.0152$	0.0177	0.0720	35.0
$(1,0.1)$	300	0.0214	$-0.0387$	0.0181	0.0770	40.7
$(4,0.1)$	300	0.0246	$-0.0355$	0.0173	0.0842	40.3
$(1,0.025)$	300	0.0308	$-0.0194$	0.0214	0.0765	34.3
$(1,0.4)$	300	0.0379	$-0.0383$	0.0317	0.1028	37.3

Appendix

Table 26: Distribution of Align–Cross gradient cosines. Quantiles and negative frequencies use valid observations pooled across three sources; zero-norm frequencies use all observations.

Weights	$\\|\nabla\mathcal{L}_{\rm align}\\|$	$\\|\nabla\mathcal{L}_{\rm cross}\\|$	Source 0	Source 1	Source 2	Early	Middle	Late
$(0.25,0.1)$	5.9381	0.0086	0.0327	0.0328	0.0135	0.0378	0.0326	0.0092
$(1,0.1)$	6.3315	0.0428	0.0208	0.0343	0.0092	0.0270	0.0286	0.0089
$(4,0.1)$	5.4435	0.0800	0.0299	0.0182	0.0256	0.0273	0.0260	0.0205
$(1,0.025)$	6.9317	0.0501	0.0393	0.0169	0.0363	0.0469	0.0307	0.0153
$(1,0.4)$	6.6684	0.0334	0.0297	0.0455	0.0385	0.0152	0.0577	0.0408

Appendix

Table 27: Gradient norms and cosines across sources and training phases. Norms are unweighted medians; other columns show mean valid cosines. Repeated diagnostic observations are not independent source runs.

Figure 11: Relation semantics and objective composition organize different factors. Mean partial geometry across three sources on 400 confirmation systems. (a) Shared factors $G_{1}=m$ , $G_{2}=(m,\gamma)$ and $G_{3}=(m,\gamma,k)$ ; (b) objective components. Blocks use separate Structure references and a common color scale. Source SDs are in Tables 32 and 33 .

Input readout ( $R^{2}$ )	Mass	Drag	Stiffness
Rendered history	$0.270\,\text{$ \pm,0.026 $}$	$0.574\,\text{$ \pm,0.009 $}$	$0.149\,\text{$ \pm,0.034 $}$
Privileged dynamics	0.762	0.937	0.760

Appendix

Table 28: Factor readout from rendered histories and privileged dynamics. Rendered-history entries show means and source SD.

Probe	Target	$G_{1}$ $R^{2}$	$G_{2}$ $R^{2}$	MSE contrast	95% interval
Ridge	$\log m$	0.5271	0.0709	0.1492	$[0.1231,0.1758]$
Ridge	$\gamma$	0.3075	0.9249	$-0.7704$	$[-0.8572,-0.6869]$
MLP	$\log m$	0.5272	0.0273	0.1635	$[0.1303,0.1974]$
MLP	$\gamma$	0.1291	0.9644	$-1.042$	$[-1.143,-0.9438]$

Appendix

Table 29: Both linear and nonlinear readouts show the

G_{1}

G_{2}

accessibility shift. Frozen centroid probes on validation systems. Error contrasts are

G_{2}

minus

G_{1}

with paired-system 95% intervals; positive favors

G_{1}

Figure 12: Persistent-context atlas under

G_{1}

and

G_{2}

. Each panel shows the same 400 validation systems. Each frozen source has an independent PCA projection; mass and drag rows recolor identical coordinates using common physical-value ranges. Axes are not aligned across sources. The atlas illustrates the representations; quantitative comparisons use the geometry and probes in Appendix E.1 .

Method	Drag $R^{2}$ $\uparrow$	Self h16 $\downarrow$	Correct $\downarrow$	Gap $\uparrow$
Native	$0.9932\,\text{$ \pm,0.0007 $}$	$0.0789\,\text{$ \pm,0.0356 $}$	n.a.	n.a.
Structure	$0.9935\,\text{$ \pm,0.0007 $}$	$0.0736\,\text{$ \pm,0.0404 $}$	$0.5512\,\text{$ \pm,0.0681 $}$	$0.0797\,\text{$ \pm,0.0079 $}$
Same-rollout alignment	$0.4434\,\text{$ \pm,0.0367 $}$	$0.0520\,\text{$ \pm,0.0078 $}$	$0.0542\,\text{$ \pm,0.0090 $}$	$0.00635\,\text{$ \pm,0.00104 $}$
Align	$0.9984\,\text{$ \pm,0.0001 $}$	$0.0140\,\text{$ \pm,0.0040 $}$	$0.0141\,\text{$ \pm,0.0041 $}$	$0.0948\,\text{$ \pm,0.0037 $}$
Align + Cross	$0.9986\,\text{$ \pm,0.0002 $}$	$0.0155\,\text{$ \pm,0.0073 $}$	$0.0155\,\text{$ \pm,0.0072 $}$	$0.0941\,\text{$ \pm,0.0031 $}$
Align–Random	$0.5299\,\text{$ \pm,0.0400 $}$	$0.0563\,\text{$ \pm,0.0100 $}$	$0.0591\,\text{$ \pm,0.0100 $}$	$0.00034\,\text{$ \pm,0.00009 $}$

Appendix

Table 30: D-Clean results on 200 one-shot sealed-test systems. Means and sample SD over three sources. Errors use each source’s own embedding target. Correct uses a same-system donor from another rollout; Gap is shuffled-minus-correct error. Native has no donor route.

Relation	Budget	Within $\downarrow$	Between $\uparrow$	Ratio $\uparrow$	Partial $\rho_{\gamma}$ $\uparrow$
Correct	R2	$78.35\,\text{$ \pm,4.67 $}$	$29.38\,\text{$ \pm,1.78 $}$	$0.375\,\text{$ \pm,0.017 $}$	$0.095\,\text{$ \pm,0.022 $}$
Correct	R4	$55.66\,\text{$ \pm,5.17 $}$	$43.32\,\text{$ \pm,2.82 $}$	$0.781\,\text{$ \pm,0.062 $}$	$0.361\,\text{$ \pm,0.036 $}$
Correct	R8	$28.09\,\text{$ \pm,1.93 $}$	$104.30\,\text{$ \pm,2.01 $}$	$3.729\,\text{$ \pm,0.313 $}$	$0.773\,\text{$ \pm,0.009 $}$
Random	R2	$70.80\,\text{$ \pm,8.74 $}$	$17.28\,\text{$ \pm,2.03 $}$	$0.244\,\text{$ \pm,0.002 $}$	$-0.001\,\text{$ \pm,0.017 $}$
Random	R4	$83.68\,\text{$ \pm,16.91 $}$	$20.39\,\text{$ \pm,3.64 $}$	$0.245\,\text{$ \pm,0.008 $}$	$-0.010\,\text{$ \pm,0.016 $}$
Random	R8	$126.90\,\text{$ \pm,0.89 $}$	$34.18\,\text{$ \pm,0.39 $}$	$0.269\,\text{$ \pm,0.004 $}$	$-0.006\,\text{$ \pm,0.004 $}$

Appendix

Table 31: PokeWorld geometry under history diversity and relation fidelity. Means and source SD over three seeds. A: validation geometry with train-standardized codes; partial drag correlation controls mass and stiffness. B: validation and frozen-confirmation results.

	Validation			Confirmation
Condition	Mass	Drag	Stiffness	Mass	Drag	Stiffness
Structure	$0.069\,\text{$ \pm,0.008 $}$	$0.194\,\text{$ \pm,0.007 $}$	$0.037\,\text{$ \pm,0.012 $}$	$0.057\,\text{$ \pm,0.010 $}$	$0.190\,\text{$ \pm,0.002 $}$	$0.018\,\text{$ \pm,0.008 $}$
$G_{1}$	$0.419\,\text{$ \pm,0.006 $}$	$0.013\,\text{$ \pm,0.009 $}$	$0.103\,\text{$ \pm,0.004 $}$	$0.489\,\text{$ \pm,0.009 $}$	$0.017\,\text{$ \pm,0.006 $}$	$0.084\,\text{$ \pm,0.003 $}$
$G_{2}$	$0.015\,\text{$ \pm,0.000 $}$	$0.720\,\text{$ \pm,0.009 $}$	$-0.013\,\text{$ \pm,0.001 $}$	$-0.004\,\text{$ \pm,0.001 $}$	$0.732\,\text{$ \pm,0.012 $}$	$-0.012\,\text{$ \pm,0.001 $}$
$G_{3}$	$0.064\,\text{$ \pm,0.013 $}$	$0.427\,\text{$ \pm,0.087 $}$	$0.067\,\text{$ \pm,0.035 $}$	$0.077\,\text{$ \pm,0.035 $}$	$0.413\,\text{$ \pm,0.082 $}$	$0.059\,\text{$ \pm,0.023 $}$
Random	$0.030\,\text{$ \pm,0.007 $}$	$0.036\,\text{$ \pm,0.014 $}$	$0.054\,\text{$ \pm,0.008 $}$	$0.022\,\text{$ \pm,0.009 $}$	$0.013\,\text{$ \pm,0.007 $}$	$0.044\,\text{$ \pm,0.006 $}$

Appendix

Table 32: Factor-selective geometry under relation refinement. Means and sample SD over three sources. Exact factor tuples are split-disjoint; individual factor levels are shared.

Condition	Mass $\uparrow$	Drag $\uparrow$	Stiff. $\uparrow$	Ratio $\uparrow$	h16 $\downarrow$
Structure	$0.058\,\text{$ \pm,0.005 $}$	$0.191\,\text{$ \pm,0.005 $}$	$0.070\,\text{$ \pm,0.011 $}$	$0.299\,\text{$ \pm,0.006 $}$	$0.404\,\text{$ \pm,0.033 $}$
Align	$0.012\,\text{$ \pm,0.009 $}$	$0.781\,\text{$ \pm,0.004 $}$	$0.016\,\text{$ \pm,0.005 $}$	$4.203\,\text{$ \pm,0.289 $}$	$0.386\,\text{$ \pm,0.019 $}$
Cross	$0.107\,\text{$ \pm,0.009 $}$	$0.229\,\text{$ \pm,0.009 $}$	$0.106\,\text{$ \pm,0.002 $}$	$0.362\,\text{$ \pm,0.009 $}$	$0.373\,\text{$ \pm,0.005 $}$
Align + Cross	$0.006\,\text{$ \pm,0.004 $}$	$0.788\,\text{$ \pm,0.001 $}$	$0.011\,\text{$ \pm,0.002 $}$	$4.411\,\text{$ \pm,0.142 $}$	$0.381\,\text{$ \pm,0.010 $}$

Appendix

Table 33: Objective components on 400 confirmation systems. Means and sample SD over three sources. Factor columns give partial geometry; Ratio is between/within separation. Native h16 errors use each source’s own embedding target.

Split	Relation	Eligible systems	Selected systems	Rollouts/system	Supported histories
Training	$G_{1}$	173–174	3	4	12
Training	$G_{2}$	19	3	4	12
Validation	$G_{1}$	35–36	3	4	12
Validation	$G_{2}$	3	3	4	12

Appendix

Table 34: Semantic eligibility and the fixed training support. Counts are per recipient. Eligible systems satisfy the relation; training uses donors from only three preselected systems.

	Native	Structure	Align	Cross	Align + Cross
Mean	0.1703	0.1716	0.1232	0.1675	0.1104

Appendix

Table 35: Align accounts for most of the SpringWorld gain; Cross lowers error further. Three-source means from separate cohorts: A, cold-start error on 100 development systems with one reader per source; B, frozen-source geometry on 178 systems. Ratio averages the source-wise between/within ratios.

Controlled JEPA objectives
Source	Cold start	Query mixture
Native	0.1716	0.8511
Structure	0.1711	0.8468
Align	0.1195	0.8033
Cross	0.1646	0.8443
Align + Cross	0.0920	0.7892

Appendix

Table 36: SpringWorld development references and physical readout. A: single-source standardized MSE on 100 cold-start and 178 query-mixture systems; supervised references use their own training recipes. B: training-fitted ridge

R^{2}

for log factors on 100 validation systems. These are separate from the sealed and multi-reader evaluations.

Figure 13: Shared hardness connects tactile histories across object shapes. Columns share hardness levels 0–2; rows differ in shape. Sensor changes are relative to the initial sample, with a common symmetric scale saturated at the pooled 97th percentile of absolute change.

Figure 14: Relation choice selects which tactile property becomes accessible. Means and sample SD over three sources on held-out grasps from six known configurations. (a) Cross-shape hardness accuracy; (b) shape balanced accuracy with one hardness level held out. Dashed lines mark the respective chance references,

1/3

and

0.50

Pairing	Factor	Probe $R^{2}$	Partial geometry $\rho$
Correct	Mass	$0.1630\,\text{$ \pm,0.0138 $}$	$0.0796\,\text{$ \pm,0.0105 $}$
Correct	Drag	$0.6087\,\text{$ \pm,0.0881 $}$	$0.4665\,\text{$ \pm,0.0461 $}$
Correct	Stiffness	$0.0589\,\text{$ \pm,0.0432 $}$	$0.0168\,\text{$ \pm,0.0186 $}$
Random mismatch	Mass	$0.2229\,\text{$ \pm,0.0274 $}$	$0.1954\,\text{$ \pm,0.0013 $}$
Random mismatch	Drag	$0.3987\,\text{$ \pm,0.0305 $}$	$0.3092\,\text{$ \pm,0.0043 $}$
Random mismatch	Stiffness	$-0.0938\,\text{$ \pm,0.1124 $}$	$0.0806\,\text{$ \pm,0.0024 $}$

Appendix

Table 37: Factor measurements under structured relation errors. Means and sample SD across three sources. Probe targets and partial correlations follow Appendix E.1 ’s PokeWorld protocol.

Figure 15: Structured mismatches redistribute factor organization. Mean paired-source changes from correct pairing in (a) probe

R^{2}

and (b) partial geometry

\rho

; positive indicates an increase. All mismatch rules use

\alpha=0.5

and three Align-only sources on 200 development validation systems. Mass-only mismatch lowers mass geometry without lowering its mean probe score. Source SDs are in Table 37 .

Pairing	Coverage (%)	Mass (%)	Drag (%)	Stiffness (%)	Separation
Correct	100	100	100	100	$1.0601\,\text{$ \pm,0.1093 $}$
Random mismatch	100	54.97	54.99	51.76	$0.9694\,\text{$ \pm,0.0236 $}$
Drag-only mismatch	100	100	50	100	$1.1618\,\text{$ \pm,0.1712 $}$
Mass-only mismatch	100	50	100	100	$3.8995\,\text{$ \pm,1.3415 $}$

Appendix

Table 38: Pairing quality, system separation and native prediction. Coverage is the fraction of recipients with a legal donor; factor columns give match rates, and separation is the between/within ratio. Native latent MSE uses each encoder’s own target space, with target variance in brackets. Errors and ratios show means and sample SD across three sources.

Relation	Tuple $p$	Mass	Drag	Stiffness	Coverage	Donor usage
True	100	100	100	100	100	1.000
Inferred	0.0639	11.15	13.30	3.64	100	0.974
RandomHistory	0.0431	10.04	10.01	3.50	100	0.980

Appendix

Table 39: Agreement and coverage of history-inferred relations. Match rates and coverage are percentages, averaged over three source-specific graphs. Donor usage is donor-rollout entropy normalized by

\log 8000

. True uses exact same-system relations.

Relation	Factor	Probe $R^{2}$	Partial geometry $\rho$
True	Mass	$0.1700\,\text{$ \pm,0.0154 $}$	$0.0737\,\text{$ \pm,0.0092 $}$
True	Drag	$0.6556\,\text{$ \pm,0.0791 $}$	$0.4807\,\text{$ \pm,0.0877 $}$
True	Stiffness	$0.0670\,\text{$ \pm,0.0020 $}$	$0.0314\,\text{$ \pm,0.0453 $}$
Inferred	Mass	$0.0538\,\text{$ \pm,0.0423 $}$	$0.0360\,\text{$ \pm,0.0069 $}$
Inferred	Drag	$0.1563\,\text{$ \pm,0.0316 $}$	$0.0061\,\text{$ \pm,0.0111 $}$
Inferred	Stiffness	$0.0314\,\text{$ \pm,0.0111 $}$	$0.0666\,\text{$ \pm,0.0028 $}$

Appendix

Table 40: Factor accessibility and geometry with inferred relations. Entries are means and sample SD across three sources.

Relation	Between/within	Native $H=1$	Native $H=4$	Native $H=16$
True	$1.0683\,\text{$ \pm,0.2664 $}$	$0.0721\,\text{$ \pm,0.0103 $}$	$0.1124\,\text{$ \pm,0.0042 $}$	$0.2327\,\text{$ \pm,0.0099 $}$
		[0.6195]	[0.6225]	[0.6249]
Inferred	$0.2577\,\text{$ \pm,0.0013 $}$	$0.0326\,\text{$ \pm,0.0031 $}$	$0.0743\,\text{$ \pm,0.0043 $}$	$0.2357\,\text{$ \pm,0.0051 $}$
		[0.7309]	[0.7366]	[0.7482]
RandomHistory	$0.2499\,\text{$ \pm,0.0117 $}$	$0.0121\,\text{$ \pm,0.0076 $}$	$0.0542\,\text{$ \pm,0.0144 $}$	$0.2268\,\text{$ \pm,0.0051 $}$
		[0.7405]	[0.7419]	[0.7436]

Appendix

Table 41: System separation and native prediction with inferred relations. Means and sample SD across three sources. The bracketed second line gives mean target variance. Each source defines its own latent prediction target.

Measurement	Factor	$\Delta$	Recovery (%)
Probe $R^{2}$	Mass	$-0.0373$	$-47.3\,[-135.5,1.0]$
Probe $R^{2}$	Drag	$+0.0498$	$9.1\,[-2.7,30.5]$
Probe $R^{2}$	Stiffness	$+0.0549$	$60.7\,[55.3,70.0]$
Geometry $\rho$	Mass	$+0.0104$	$21.6\,[-27.6,53.4]$
Geometry $\rho$	Drag	$-0.0281$	$-6.3\,[-8.6,-4.2]$
Geometry $\rho$	Stiffness	$+0.0343$	— a

Appendix

Table 42: Inferred-relation changes and recovery relative to True.

\Delta

is Inferred minus RandomHistory. Recovery intervals are 95% paired-source bootstrap intervals over three seeds; they do not measure cross-task stability. Dashes denote inapplicable ratios.

Figure 16: Physical donor substitutions redirect predictions and produce error valleys. (a) One preselected case: lines show simulator truth and markers show predictions at horizons 1/2/4/8/16. (b) Valley depths for four separately fitted recipes on 64 systems, with paired-system 95% intervals. Align + Cross uses the pooled intervals in Table F.1 .

Factor	Source	Valley depth	95% interval	$\rho$
Mass	Pooled	0.1820	$[0.1320,0.2343]$	0.591
	Seed 0	0.1950	$[0.1379,0.2558]$	0.606
	Seed 1	0.1612	$[0.1131,0.2134]$	0.592
	Seed 2	0.1899	$[0.1364,0.2462]$	0.601
Drag	Pooled	0.0303	$[0.0216,0.0400]$	0.585
	Seed 0	0.0334	$[0.0238,0.0439]$	0.600

Appendix

Table 43: Align + Cross yields positive donor-matching valleys in every source. Depth is off-diagonal minus diagonal h16 error on 64 development systems. Source results and recipe comparisons use three frozen sources and one reader per source. Paired-system 95% intervals are conditional on fitted models.

Figure 17: Shared-factor relations redirect predictive responses. (a) Mean

G_{1}

-minus-

G_{2}

mass and

G_{2}

-minus-

G_{1}

drag projections on 144 systems. (b)–(c) Wrong-factor minus matched h16 object-state MSE on a separate 96-system bank. Pointwise 95% intervals resample systems with three sources fixed (and state decoders fixed in (b)–(c)). Yellow marks an interval above zero. Native projections and physical MSE have distinct units.

Figure 18: Cross reduces all five matched-donor forecasting endpoint means. (a) RH20T frames and synchronized force/TCP observations. (b) Cross-Indep minus Structure on 25 held-out tasks; negative favors Cross. Force/torque uses normalized MSE and TCP uses cm. Task-paired 95% intervals follow averaging over three sources. Absolute risks on the separate 24-task bank are in Table 45 .

	Force/torque MSE			TCP (cm)
Condition	h1	h4	h16	h4	h16
Native	0.6346	1.117	1.497	1.962	4.063
Structure	0.6199	1.099	1.498	1.944	3.965
Align–Indep	0.6633	1.132	1.503	2.262	4.429
Align–Random	0.6232	1.106	1.510	2.045	4.257
Cross–SameEp	0.6145	1.090	1.495	1.963	3.965

Appendix

Table 44: Self-context and matched-donor forecasting on RH20T. Equal-weight means over 25 held-out tasks and three sources. Force/torque uses normalized MSE; TCP uses cm. LD denotes LowDim; A+C denotes Align + Cross. Dashes indicate unavailable matched inputs. Shading highlights the Structure/Cross comparison.

Endpoint	$S_{\mathrm{A+C}}$	$S_{\rm Mono}$	$D_{\rm spec}$
FT h4 MSE	$0.00260\;[0.00107,0.00429]$	$0.00030\;[-0.00050,0.00134]$	$\mathbf{0.00230}\;[0.00073,0.00404]$
TCP h16 (cm)	$0.0891\;[0.0209,0.1614]$	$0.0176\;[-0.0025,0.0414]$	$0.0715\;[-0.0042,0.1480]$

Appendix

Table 45: Donor specificity and absolute risk on the 24-task bank. Means weight tasks equally after averaging three sources, with task-paired 95% intervals. Specificity is hard-minus-matched error; contrasts are A+C minus Monolithic. Positive risk differences favor Monolithic. FT h4 is primary; TCP h16 is secondary.

Intervention/model	Metric	Difference	95% task interval
A+C: action zero	FT h4	0.002074	$[0.000740,0.003650]$
A+C: action zero	TCP h4	0.06439	$[0.04507,0.08274]$
A+C: action zero	TCP h16	0.1399	$[0.07687,0.2025]$
Align + Cross	FT h4	0.009692	$[0.004828,0.01624]$
Align + Cross	TCP h4	0.4705	$[0.2801,0.6948]$
Align + Cross	TCP h16	0.3385	$[0.1768,0.5304]$

Appendix

Table 46: Action use and donor-count sensitivity on a separate 25-task bank. A: action-zero minus observed, and shuffled-minus-correct donor effects. B: random-minus-correct effects at fixed donor count

K

and fixed weights. Pointwise 95% task-bootstrap intervals follow averaging three sources. FT uses MSE; TCP uses cm.

Model	$K$	FT h4 MSE	TCP h4 (cm)	TCP h16 (cm)
Align + Cross	1	$0.01019$ $[0.00476,0.01786]$	$0.4686$ $[0.2741,0.6968]$	$0.3368$ $[0.1719,0.5327]$
Align + Cross	2	$0.00874$ $[0.00325,0.01648]$	$0.3652$ $[0.1675,0.6004]$	$0.2614$ $[0.09066,0.4618]$
Align + Cross	4	$0.00794$ $[0.00240,0.01566]$	$0.2978$ $[0.08930,0.5383]$	$0.2191$ $[0.04070,0.4249]$
Cross	1	$0.01606$ $[0.00836,0.02703]$	$0.00488$ $[-0.01979,0.03002]$	$0.01573$ $[-0.04830,0.07455]$
Cross	2	$0.01593$ $[0.00634,0.02960]$	$0.00691$ $[-0.01343,0.02892]$	$0.01599$ $[-0.02912,0.05778]$
Cross	4	$0.01602$ $[0.00531,0.03114]$	$0.00869$ $[-0.01117,0.02953]$	$0.01166$ $[-0.02378,0.04752]$

Appendix

Table 66

Figure 19: Restitution dominates reader-memory accessibility and donor effects. (a) Training-fitted ridge

R^{2}

on RSSM reader memory. (b) Wrong-one-factor minus matched MSE on 2,000 recipients through fixed weights. Bars average three source/reader fits; symbols show individual fits. SPRII uses Cross.

Recipe	Repeat	S3	S5	S8
SPRII (Cross)	1.29	1.171	1.156	1.147
Native	1.532	1.286	1.241	1.206
Random	1.497	1.356	1.333	1.316
Structure	1.517	1.288	1.246	1.218

Appendix

Table 47: Distinct histories lower error through the fixed S3-trained RSSM reader. MSE averages three fits; Repeat averages repetitions of each individual history. Reader weights remain fixed.

Recipe	Factor	Source 1	Source 2	Source 3
SPRII (A+C)	Mass	0.0509	0.0546	0.0678
	Friction	0.0027	$-0.0020$	0.0002
	Restitution	0.8297	0.8266	0.8210
Native	Mass	0.0283	0.0271	0.0411
	Friction	$-0.0014$	0.0024	0.0009
	Restitution	0.7581	0.7599	0.7684

Appendix

Table 48: Reader-memory probes, RSSM geometry and separate donor assays. A: training-fitted CoPhyNet memory probes, including the learned projection. B: RSSM same-physics/wrong-physics code-distance ratios. C: separate single-source evaluations, reporting wrong-factor minus matched MSE. These panels measure distinct channels and cohorts.

Frozen base	Gain	95% interval	Positive systems
Persistent-JEPA	5.329	$[3.930,6.724]$	65.04%
Masked-GRU (64101)	6.930	$[5.415,8.621]$	67.97%
Masked-GRU (64103)	6.661	$[4.878,8.486]$	68.36%

Appendix

Table 49: Recipient-specific updates improve all three frozen Swimmer predictors. Gain is wrong-system minus own-system update error in

10^{-5}

units. Intervals are paired-system 95% bootstrap intervals on 512 new systems.

Horizon	Cold: gain [95% interval]	Moving: gain [95% interval]
1	$1.95\,[-8.47,11.01]$	$1.15\,[-0.83,3.01]$
2	$13.39\,[3.93,21.49]$	$1.98\,[-0.22,4.02]$
4	$23.19\,[14.37,30.35]$	$3.23\,[0.91,5.77]$
8	$35.20\,[26.74,41.92]$	$11.62\,[7.17,16.57]$
16	$35.68\,[28.22,42.27]$	$29.73\,[22.68,36.45]$

Appendix

Table 50: History benefit increases with prediction horizon. Percentage error reduction of separately fitted Persistent versus Null readers, with pointwise stratified system-bootstrap 95% intervals.

Figure 20: Direct context outperforms the physical bottleneck across D-Clean horizons. (a) Raw-state MSE (log scale); Fixed Wrong substitutes donors through the Persistent reader. (b) Decode-minus-Persistent MSE with pointwise system-bootstrap 95% intervals, conditional on fitted sources and readers. The primary endpoint is h32.

Horizon	Null	Persistent	Decode	Oracle	Fixed Wrong
1	0.000577	0.000011	0.000013	0.000007	0.003227
2	0.001821	0.000020	0.000023	0.000009	0.009263
4	0.005689	0.000048	0.000056	0.000014	0.025560
8	0.015320	0.000120	0.000149	0.000031	0.061030
16	0.031970	0.000278	0.000349	0.000078	0.110300
32	0.059360	0.000590	0.000764	0.000223	0.180900

Appendix

Table 51: Direct-context and bottleneck readers use the same D-Clean donors. (A) Raw-state MSE; (B) pointwise paired-system 95% intervals conditional on the fitted source–reader grid; (C) independent validation of the frozen drag decoder. The primary endpoint is h32.

Population	Null	Persistent	Decode	Oracle
Cold start: 100 systems / 600 cases	0.17780	0.11360	0.11130	0.06188
Full mixture: 178 systems	0.87170	0.80970	0.80870	0.76590

Appendix

Table 52: SpringWorld reader error and donor-factor accessibility. (A) Common-adapter MSE across three sources and three readers. (B) Training-fitted log-parameter probes on 712 donor histories from 178 validation systems, weighted equally by system.

Figure 21: Persistent context improves object-state prediction, with mixed effects on finger state. Null-minus-Persistent MSE on the development bank; positive favors Persistent. Pointwise paired-system 95% intervals condition on the fitted source–reader grid; the aggregate interval adjusts for three contrasts. Components are not additive.

Arm	MSE
Null	0.9629
Persistent	0.9624
Shuffled	0.9637
Oracle	0.961

Appendix

Table 53: Aggregate PokeWorld gains are small despite component-level benefits. Separately fitted readers on 400 development systems; aggregate 95% intervals adjust for three contrasts. Oracle receives true parameters.

A. Continuous slope of $\Delta U$ against sensitivity dominance
Source seed	0	1	2
Expanded-grid slope	0.003838	0.003307	0.001284
Mean: $0.002810$ ; 95% system-bootstrap interval: $[-0.004443,0.009203]$

Appendix

Table 54: PokeWorld factor-targeted utility and donor accessibility. (A) The

G_{1}

-minus-

G_{2}

history-benefit slope against simulator sensitivity has a system-bootstrap interval spanning zero. (B) Probe

R^{2}

on actual donor windows, averaged across source seeds. The development grid includes the initial fits.

Figure 22: Reader recovery depends on source family and donor context. (a) Mean within-fit error reduction,

100[E(M_{1})-E(M_{2})]/E(M_{1})

; positive favors

M_{2}

. Reader percentages are averaged within source, then across sources. (b) SpringWorld raw MSE under own- and cross-system donors through fixed

M_{2}

readers;

s=0

gives the recipient-only base

R_{0}

Family	Sources $\times$ readers	$M_{1}$	$M_{2}$	Gain	Weighted gain
Align + Cross	$3\times 3$	0.6185	0.5695	0.04901	0.05059
Structure	$1\times 3$	0.6316	0.6378	$-0.006213$	$-0.006971$
Poke $G_{1}$	$1\times 2$	0.8099	0.8127	$-0.002805$	$-0.001495$
Poke $G_{2}$	$1\times 3$	0.7919	0.7987	$-0.006841$	$-0.006443$

Appendix

Table 55: Reader recovery differs across frozen source families. Descriptive mean errors and gains

E(M_{1})-E(M_{2})

; positive favors

M_{2}

. Weighted gains use the separate physically weighted route. These fitted-run means do not estimate uncertainty.

Endpoint	$M_{1}$	$M_{2}$ matched	$M_{2}$ mismatched
Full development	$0.6185\,\text{$ \pm,0.009762 $}$	$0.5695\,\text{$ \pm,0.01796 $}$	$0.6240\,\text{$ \pm,0.007561 $}$
Cold start, h16	$0.1799\,\text{$ \pm,0.006272 $}$	$0.1216\,\text{$ \pm,0.01436 $}$	$0.1835\,\text{$ \pm,0.008084 $}$

Appendix

Table 56: Matched context lowers error with identical

M_{2}

architecture. Task MSE, mean

\pm

sample SD across three sources, each averaging three readers. Both populations are development data: 178 systems for the full mixture and 100 for cold-start h16.

Figure 23: Paired error differences for the matched-architecture control. (a) Full development mixture; (b) cold-start h16. Negative values favor the first reader (

10^{-3}

MSE). Source-paired 95%

t

intervals use three source means; hierarchical 95% intervals resample sources, readers and systems. Both condition on development-selected fits; matched and mismatched

M_{2}

readers are fitted separately.

$\alpha$	B/W	$R^{2}_{\log\gamma}$	1k gain	5k gain	5k 95% interval
0	0.320	0.101	$-2.512$	$-1.890$	$[-8.366,+3.116]$
0.5	1.042	0.254	$+0.061$	$-2.942$	$[-8.255,+1.948]$
1	3.796	0.797	$-4.469$	$-6.291$	$[-12.010,-0.328]$

Appendix

Table 57: Stronger formation does not ensure greater reader value. h16 gain is

E(M_{1})-E(M_{2})

10^{-3}

normalized MSE; positive favors

M_{2}

. Budgets share frozen sources and development systems, averaging three readers per source. B/W uses training-standardized latent statistics.

A. Standard $M_{2}$ ; displayed in Figure 22
Source	Donor	$s=0$	$0.25$	$0.5$	$0.75$	$1$
Align + Cross	Own	0.6333	0.6074	0.5851	0.5703	0.5695
	Cross-system	0.6333	0.6339	0.6450	0.6700	0.7125
Structure	Own	0.6419	0.6384	0.6363	0.6364	0.6378
	Cross-system	0.6419	0.6394	0.6381	0.6386	0.6404
B. Physically weighted $M_{2,\mathrm{phys}}$

Appendix

Table 58: Donor identity changes the response to context scale. Development MSE as persistent inputs are multiplied by

s

, with donor substitutions through fixed readers. At

s=0

, the route equals the recipient-only base

R_{0}

, not

M_{1}

Source	$M_{2}$	$M_{1}$	Recipient-only
R8	0.9011	0.9026	0.9118
$G_{2}$	0.9132	0.9085	0.9080

Appendix

Table 59: Object-only PokeWorld reader comparisons. h16 object-state MSE on 400 development systems for one source–reader pair.

M_{1}

and

M_{2}

are selected independently; the paired interval is conditional on these fits and evaluation systems.

Group	CaDM	Random relation	SPRII (R7)
ID	$-326.5\,\text{$ \pm,46.5 $}$ (96.7%)	$-321.4\,\text{$ \pm,33.2 $}$ (100.0%)	$-323.9\,\text{$ \pm,26.2 $}$ (100.0%)
OOD_c0	$-376.6\,\text{$ \pm,63.4 $}$ (93.3%)	$-346.8\,\text{$ \pm,30.6 $}$ (100.0%)	$-322.2\,\text{$ \pm,29.4 $}$ (100.0%)
OOD_c1	$-387.0\,\text{$ \pm,11.9 $}$ (93.3%)	$-400.1\,\text{$ \pm,24.6 $}$ (93.3%)	$-381.0\,\text{$ \pm,13.6 $}$ (100.0%)
OOD_c2	$-355.5\,\text{$ \pm,33.9 $}$ (90.0%)	$-479.6\,\text{$ \pm,126.7 $}$ (76.7%)	$-370.2\,\text{$ \pm,51.8 $}$ (90.0%)
OOD_c3	$-1100\,\text{$ \pm,273.5 $}$ (20.0%)	$-1185\,\text{$ \pm,131.2 $}$ (6.7%)	$-1060\,\text{$ \pm,384.4 $}$ (30.0%)

Appendix

Table 60: Pendulum control with one ID-selected R7 recipe. Mean return

\pm

sample SD across three seeds, with success percentage in parentheses; higher is better. The same recipe is used for all OOD groups.

Figure 24: CoPhy gains depend on learner, scene and comparator. (a)–(c) Percentage MSE reduction against Native, Structure and Random:

100(\mathrm{control}-\mathrm{SPRII})/\mathrm{control}

; positive favors SPRII. These are single-source development comparisons. CoPhyNet Structure was not evaluated in this matrix; held-out Collision results appear separately in Table 10 .

Cross: self-supervised dynamics objectives
Scene	Family	Native	Structure	SPRII	Random
Balls	JEPA	1.2563	1.2630	1.2396	1.5827
	CPC	1.4399	1.4239	1.5042	1.6228
	RSSM	1.3105	1.2582	1.1254	1.3155
Collision	JEPA	0.2512	0.2212	0.2297	0.2717
	CPC	0.2448	0.2283	0.1778	0.2220

Appendix

Table 61: CoPhy prediction error by learner and scene. Source-0 development MSE under S3/query3 and source100/head100. Dashes mark unevaluated Structure arms; bold marks the lowest MSE per row.

	Prediction MSE			Support $R^{2}_{S}$			Reader memory $R^{2}_{M}$
Arm	Src 0	Src 1	Src 2	Src 0	Src 1	Src 2	Src 0	Src 1	Src 2
JEPA / Balls
Native	1.256	1.319	—	0.7977	0.7981	—	0.7865	0.7778	—
Structure	1.263	1.294	—	0.7855	0.7866	—	0.7832	0.7788	—
Cross	1.240	1.305	—	0.9162	0.9185	—	0.9051	0.9116	—
Random	1.583	1.644	—	0.6777	0.6918	—	0.6884	0.6823	—

Appendix

Table 62: CoPhy prediction and accessibility across source seeds. Development results include reader-selection examples.

S

denotes independent support and

M

fitted reader memory. Probe targets are restitution (Balls), mass (Collision) and vertical gravity (Blocktower). Dashes mark unevaluated cells.

A. Absolute risks under recorded context conditions
Family	Arm	Matched	Null	Wrong
RSSM	SPRII (C)	1.171	1.892	2.296
	Native	1.286	2.485	2.293
	Random	1.356	2.004	2.300
	Structure	1.288	1.937	2.335
CoPhyNet	SPRII (A+C)	1.044	1.791	2.228

Appendix

Table 63: All five Balls contrasts favor relation training. Three-source means on 2,000 development recipients. Recipient-bootstrap 95% intervals are Bonferroni-adjusted over five contrasts, conditional on fitted models and excluding selection uncertainty.

Reader recipe	Balls	Collision	Blocktower
Main initialization, dropout 0.1, initial	$-5.448$	$-4.089$	$-1.224$
Alternative initialization, dropout 0	$+7.506$	$-2.104$	$-1.477$
Alternative initialization, dropout 0.1	$+6.074$	$-4.426$	$+2.090$
New initialization, dropout 0	$+0.617$	$-7.591$	$+0.621$
New initialization, dropout 0.1, every step	$+6.646$	$-2.925$	$+1.134$

Appendix

Table 64: CoPhyNet gains depend on the reader recipe. SPRII versus Native error reduction (%) for one frozen source on validation data (source-50/head-100).

Configuration	Coverage	Numerical comparison	Finding / scope
JEPA Collision: J2	1,994 test episodes; 3 seeds per arm	J2 $0.184\,\text{$ \pm,0.005 $}$ ; Structure $0.215\,\text{$ \pm,0.007 $}$	14.2% lower than Structure (all arms source-50); all four arms in Table 10 .

CoPhyNet Collision: Cross	4,000 development recipients; seed 0	Cross 0.2032; SPRII S1 reference 0.2107	Near Native; paired interval against Native includes zero.

CPC Balls: Align	2,000 development recipients; seed 0	Align 1.393; SPRII C2 reference 1.415; Structure 1.388	Lower than the SPRII C2 reference, higher than Structure.

Appendix

Table 65: CoPhy operation and configuration comparisons. Errors are lower-is-better MSE. Development rows use one fitted source/reader seed; paired intervals condition on the reported models and evaluation units.

History	Familiar probe MSE	Familiar return	Held-out dev. return
25%	$-0.00567\,\text{$ \pm,0.00810 $}$	$1.489\,\text{$ \pm,2.335 $}$	$-2.222\,\text{$ \pm,15.21 $}$
50%	$-0.01344\,\text{$ \pm,0.00728 $}$	$0.389\,\text{$ \pm,1.110 $}$	$18.44\,\text{$ \pm,20.66 $}$
100%	$\mathbf{-0.02120}\,\text{$ \pm,0.00610 $}$	$\mathbf{1.889}\,\text{$ \pm,2.175 $}$	$\mathbf{-1.111}\,\text{$ \pm,11.44 $}$

Appendix

Table 66: Full-history alignment improves familiar-partner returns in all three seeds. Paired Align

-

H+VC differences, mean

\pm

sample SD across three training seeds. Lower probe MSE and higher return are better. Familiar and held-out development partners are reported separately.

History	Recipe	Familiar MSE $\downarrow$	Held-out dev. MSE $\downarrow$	Distance ratio
25%	VC	$0.06788\,\text{$ \pm,0.00667 $}$	$0.13880\,\text{$ \pm,0.04230 $}$	$1.343\,\text{$ \pm,0.061 $}$
25%	A	$0.06220\,\text{$ \pm,0.00974 $}$	$0.09733\,\text{$ \pm,0.01384 $}$	$1.917\,\text{$ \pm,0.071 $}$
50%	VC	$0.06934\,\text{$ \pm,0.00811 $}$	$0.14090\,\text{$ \pm,0.02324 $}$	$1.408\,\text{$ \pm,0.019 $}$
50%	A	$0.05590\,\text{$ \pm,0.00736 $}$	$0.08966\,\text{$ \pm,0.01240 $}$	$2.329\,\text{$ \pm,0.326 $}$
100%	VC	$0.07516\,\text{$ \pm,0.00791 $}$	$0.12400\,\text{$ \pm,0.02857 $}$	$1.458\,\text{$ \pm,0.036 $}$
100%	A	$0.05395\,\text{$ \pm,0.01106 $}$	$0.10630\,\text{$ \pm,0.01046 $}$	$2.269\,\text{$ \pm,0.206 $}$

Appendix

Table 67: Overcooked behavioral readout and cooperation returns. Mean

\pm

sample SD across three training seeds. A denotes Align; VC denotes H+VC. The distance ratio is different-/same-partner distance on the familiar behavioral-signature bank, computed per seed before averaging. Random has no return evaluation.

Figure 25: Additional histories narrow the FHN endpoint gap on all three splits. H50 relative-

L_{2}

error: (a) ID, (b) OOD-Intra, (c) OOD-Extra. Points and whiskers show mean

\pm

sample SD across three sources. Annotations give

100(\mathrm{SPRII}-\mathrm{NOD})/\mathrm{NOD}

K=1,4

	$K=1\to 4$ reduction		NOD error		SPRII error
Split	SPRII	NOD	$K=1$	$K=4$	$K=1$	$K=4$
ID	16.9%	8.4%	0.02388	0.02188	0.02728	0.02266
OOD-Intra	18.0%	9.2%	0.02336	0.02121	0.02627	0.02155
OOD-Extra	10.9%	7.1%	0.03233	0.03002	0.03418	0.03046

Appendix

Table 68: Additional histories narrow the FHN prediction gap. H50 relative-

L_{2}

error averaged across three sources; lower is better. Both methods receive the same

K

independent histories, with different training budgets as described above.

Split	Method	Src 42	Src 43	Src 44
ID	NOD	0.9698	0.9301	0.9836
ID	SPRII	0.9444	0.3775	0.8952
Intra	NOD	0.9269	0.8098	0.9377
Intra	SPRII	0.8602	0.5513	0.6923
Extra	NOD	0.9351	0.9139	0.9284
Extra	SPRII	0.8587	0.0949	0.8730

Appendix

Table 69: FHN accessibility varies by factor, split and source. Training-fitted probe

R^{2}

with one history (

K=1

Figure 26: A broader sharing relation can select a different predictive direction. In the Proposition 3 example, broadening (a)

G_{1}

to (b)

G_{2}

changes the selected direction (diamond) from mass to drag. Gray bars show predictive scores; colored bars subtract relation disagreement.

Work	Relevant construction	Connection developed by SPRII
FCRL	Same-function observation sets; contrastive encoding; new decoders on a fixed encoder.	Relation training within predictive/action learners; selected shared factors; separate measurement of code access and predictor reliance.

NOD	A separate same-system trajectory supplies a low-dimensional code for state evolution.	Embedding, outcome, and action targets; Align/Cross components; controlled relation semantics and information conditions.

RIA	Predictive interventions estimate relations used to organize dynamics context.	Observed relations provide a controlled variable whose reliability and semantics can be changed independently.

Appendix

Table 70: Closest methodological connections.

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