GenCoord: Executable Commitments for Embodied LLM Agents under Private Information

Junning Zhu; Peng He; Haohan Yuan; Jianpeng Liang

cs.AIAug 22, 2026

GenCoord: Executable Commitments for Embodied LLM Agents under Private Information

Authors: Junning Zhu, Peng He, Haohan Yuan, Jianpeng Liang

Organizations: Beijing Normal-Hong Kong Baptist University · Tsinghua University · University of North Carolina at Charlotte · University of California San Diego

Abstract

Embodied language-model agents must coordinate interdependent actions while each holds private task information. In dialogue-based coordination, models generate messages and interpret replies to decide what each agent should do, spending inference before the team acts. We propose GenCoord, a decentralized protocol that makes the division of work itself the message. A proposal pairs the sender's skill path with its partner's continuation in Short DSL, a compact language for expressing joint work. Compatible paths form commitments linking the sender's delivery to the partner's work after verified receipt. We check fulfillment separately from original-task success. Complete requests also let a deterministic receiver proceed without another model call. In Minecraft, trained local 0.8B agents match the task success of controlled free-form communication while reducing median commitment time by 68.2%. The same-field full-key JSON control approaches free-form output length and delay, indicating that the amount generated matters even when messages are structured. With the same prompted DeepSeek model, both protocols complete nearly all tasks, but dialogue generates 2.8 times as many output tokens. Multistep commitments improve held-out-template success from 91.3% to 98.1% with fewer decisions. These results point to a design principle for embodied cooperation: make the division of work the message, and spend inference on choosing and revising the work.

Figures & tables

Figure 1. GenCoord in the shared-graph configuration: two information directions create executable obligations. SELF+REQ pairs the sender’s work with the peer’s continuation. The blue route realizes and resolves the delivered request; the red route applies a deterministic response policy, followed by requester revision and resolution. The example shows how the task-prescribed processing mode changes the handoff. The sender owes delivery, and the receiver owes completion conditional on verified receipt. Checked execution verifies fulfillment through the handoff and declared outcome. An overview with two agents at the top, a blue forward route and a red feedback route in the middle, and checked execution at the bottom. Agent A's private fact selects the request in a SELF+REQ proposal. SELF describes the sender's path and REQ the peer's continuation, encoded with the Short DSL precedence separator >. Agent B observes a task-prescribed processing mode: B crafts from planks delivered by A, or A crafts and B receives the table. A shared task graph and primitive semantics support both routes. In the blue route, receiver realization uses local context x sub j, the actual delivered request r delivered, and executable prior K sub j to produce continuation u sub j. It can be learned or deterministic. The path resolver checks a match to the normalized delivered request, agreement of handoff fields, and an acyclic dependency merge, producing the forward commitment C forward. In the red route, a deterministic policy maps the requested suffix and processing mode to ACCEPT, REJECT, or COUNTER(alpha). ACCEPT and COUNTER lead to requester revision and resolution, producing C feedback; REJECT creates no commitment. The example table assigns B to craft after receiving four planks under ACCEPT, or A to craft and hand over the table under COUNTER; both use the order chest as the destination. Either route yields the resolved obligation: A owes delivery and B owes completion after verified receipt. The bottom strip checks typed fields, materializes the dispatch plan and compiles it to skills, executes world actions and the handoff, and verifies receipt and declared completion.

Figure 2. Making an obligation observable. A resolved object is checked, materialized as a dispatch plan, and executed through Minecraft skills. Inventory readback verifies that the receiver obtained the planks and that the crafting table reached the order chest. These checks establish fulfillment of the commitment. The resolved SELF+REQ object assigns acquisition and delivery to Agent A, followed by crafting and deposit by Agent B. Checked task fields determine a dispatch plan and Mineflayer calls. Four oak planks transferred to the receiver establish handoff; one crafting table in the order chest establishes the declared terminal predicate. Original-task success is evaluated against its own fixed specification.

Figure 3. Four families vary the executable consequence of a private fact: destination, post-handoff transformation and output, assigned remaining work, or downstream continuation. Both Recipe templates transfer oak planks. The Allocation alternatives change storage/build work or the assigned wall/roof component. Four paired-task panels identify the changed request fields. Destination selects a terminal location. Recipe selects a post-handoff transformation and output with the same oak-plank handoff. Allocation selects assigned remaining work, including storage versus building or wall versus roof. Active Branch selects the downstream continuation. These alternatives produce distinct task-relevant physical outcomes.

Figure 4. Communication costs relative to Short DSL and commitment horizon. (a) Local JSON and controlled free-form have similar output-token and median-TTC ratios, and all three complete the quality pool. (b) DeepSeek dialogue’s output-token ratio is larger than its median-TTC ratio. Bars in (a,b) share a scale, with Short DSL at 1. Panel (a) uses the 40-cluster timing sample; panel (b) combines quality-pool output means with medians from 64 timing trials per protocol. (c,d) Multistep commitments improve held-out-template success and reduce online decisions. Four panels compare normalized communication costs and commitment horizons. Panels a and b share a vertical scale relative to Short DSL at one. Green solid bars indicate Short DSL, light blue bars indicate JSON, and gray hatched bars indicate the natural-language baseline. In panel a, JSON and controlled free-form communication each produce about 3.4 times the output tokens and take about 3.1 times the median commitment time of Short DSL. All three complete the 128-cluster quality pool; the plotted costs come from the 40-cluster timing sample. In panel b, dialogue with the same DeepSeek model produces 2.8 times the output tokens and takes 1.3 times the median commitment time. Quality-pool success is 100 percent for Short DSL and 99.7 percent for dialogue. Output means use that quality pool, while timing medians use 64 interleaved trials per protocol. Panel c shows overall and per-family held-out-template success for single-step and multistep control, including 91.3 versus 98.1 percent overall. Panel d shows 2.91 versus 1.98 mean online model decisions, a reduction of 32 percent.

Surface	Success	Calls	Input tok.	Output tok.	Wire (B)	TTC p50/p95 (s)	Episode p50/p95 (s)
Short DSL	128/128	2.00	949.1	78.6	76.3	2.27 / 2.62	24.86 / 27.06
JSON	128/128	2.00	1154.7	267.6	303.4	7.07 / 7.86	29.94 / 31.82
Controlled free-form	128/128	2.50	1585.2	265.8	1061.1	7.13 / 7.86	30.09 / 31.92

Table 1. Communication surfaces at matched task success. Quality and episode time use the 128-cluster pool; online costs use a 40-cluster surface-comparison timing sample. Wire is measured peer payload; TTC ends at commitment formation. p50/p95 denote median/95th percentile.

Metric	Short DSL	Dialogue
Task success	768/768	766/768
Calls	2.00	2.12
Peer payload (B)	112.75	913.76
Output tokens	100.4	276.3
Mean TTC (s)	4.29	6.17
Median TTC (s)	4.38	5.52

Table 2. Communication protocols with the same DeepSeek model. Quality uses 768 trials per protocol; TTC uses 64 interleaved trials.

Selector	Accuracy (%)	Time (ms)
Embedding retrieval	82.19	92
Factor (local 0.8B)	95.94	412
DeepSeek	99.97	1,402
Retrieval + DeepSeek	88.91	350

Table 3. Selecting an assignment from new descriptions. Accuracy is measured before execution on 640 descriptions (3,840 choices for DeepSeek). Mean decision time comes from 128 timed decisions per strategy and includes service and network time for API calls.

Appendix figures & tables58 assets

Supplementary material from the paper’s appendix.

Appendix

Figure S1. How a proposal becomes executable joint work. In the forward route (a), the receiver realizes a delivered request through generation or deterministic instantiation before the resolver checks compatibility and dependencies. In the feedback route (b), the deterministic response selects acceptance, a counteroffer, or rejection, and the requester revises an accepted or countered proposal. Physical checks evaluate receipt H and declared completion T, distinguishing fulfillment F from original-task success G. Panel (c) locates model calls and checks across implementations. Two swimlane diagrams and a table. In the forward request, the sender forms SELF plus REQ in learned call 1, the delivered request can be substituted by a legal alternative, the receiver realizes it in learned call 2 or deterministically, the path resolver checks match, handoff, and acyclicity, and the resolved commitment is executed, followed by the handoff check H and the terminal check T. Fulfillment F is scored against the delivered obligation and task success G against the original task. In feedback, a deterministic peer policy returns ACCEPT, COUNTER, or REJECT; the response can be substituted; the requester revises in a second learned call when its initial proposal equals the canonical one; REJECT creates no commitment. A table lists learned calls and checks for five configurations.

Symbol	Meaning and domain
$\mathcal{A}=\{a_{i},a_{j}\}$	the two agents; protocol roles are written as snd , rcv , req , and peer
$\mathcal{K}=(\mathcal{S},\mathcal{T},\Theta,\Gamma)$	core executable prior: skills $\mathcal{S}$ , shared task DAG $\mathcal{T}$ , schema $\Theta$ , and grounding/checker contract $\Gamma$ ; in the private-structure setting only primitive semantics $\mathcal{K}_{0}$ are shared
$\xi_{\mathrm{pub}}$ , $\omega_{t}$ , $x_{i}^{t}$	public episode instantiation, current executable world state, and agent $i$ ’s model-visible local context at round $t$ ; $\omega_{t}$ is visible only to the runtime verifier
$\hat{y}_{\mathrm{snd}},q_{\mathrm{snd}}$ ; $\hat{y}_{\mathrm{rcv}},q_{\mathrm{rcv}}$	raw model strings and parsed sender/receiver proposals; $q=\bot$ denotes parse failure
$r_{\mathrm{snd}\rightarrow\mathrm{rcv}}$ , $u_{\mathrm{rcv}}$	requested peer path and receiver-local realization suffix
$c_{\mathrm{peer}}$ , $\rho$ , $\alpha$	peer-local processing mode (data field capability ), bounded response, and selected counter_offer_id

Appendix

Table S1. Notation for proposals, resolution, dispatch, and world evaluation in Algorithms S1 – S2 .

Decision / stage	Valid fields or condition	Protocol consequence
Accept	reason code; no counter_offer_id	second requester call preserves the normalized initial branch
$\textsc{Counter}(\alpha)$	reason code and exactly one admissible counter_offer_id = $\alpha$	second requester call realizes the selected bounded branch
Reject	reason code; no counter alternative	creates no commitment; no second generation or dispatch
Invalid response	unknown decision, forbidden field, missing alternative, or inadmissible ID	CONTRACT_REJECT
Parse	raw string cannot be decoded	PARSE_FAILURE
Contract	schema, role, skill, argument, target, or response check fails	CONTRACT_REJECT

Appendix

Table S2. Response options and validation outcomes. Each option identifies the information returned to the requester and the action permitted by the protocol.

Stage	Check	Accepted condition
Pre-materialization	Syntax and identity	two-line grammar; actors and peer match the role instance
	Skill and arguments	catalog path, required keys, types, and quantities are valid
	Binding and handoff	branch is admissible; item, quantity, sender, and receiver agree
	Dependency merge	selected predecessor graph is complete and acyclic
Post-materialization	Static plan consistency	resolved actors, fields, handoff edge, and checker attachment match the validated commitment
Post-execution	World verification	handoff event and declared terminal predicate both pass

Appendix

Table S3. Validation and physical verification through the execution pipeline. The checks establish an admissible commitment before evaluating transfer and completion.

Figure S2. A complete Short DSL episode. Panel (a) follows sender B’s proposal and receiver A’s realization to commitment formation at 2.21 s. Panel (b) shows server-read inventories before and after transfer and at completion, omitting tool items. Panel (c) aligns the two generation calls with thirteen dispatched skills and inventory readbacks, with dashed lines locating the handoff H and terminal T checks. Three panels for one logged episode. Panel a shows the sender and receiver Short DSL records and the resolved commitment. Panel b shows inventory slots before the handoff, after the handoff, and at the terminal check: B holds eight oak planks, then A holds eight, then A holds one crafting table and four planks. Panel c shows two formation calls and thirteen dispatched skills on a time axis, with inventory counts after each dispatch and dashed lines at the handoff and terminal checks.

Family	Template	Source material	Balanced bindings	Binding-dependent request path	Changed field
Destination	Build Site A/B	oak planks	SITE_A / SITE_B	build.component	destination
Destination	Chest A/B	cobblestone	CHEST_A / CHEST_B	resource.deliver	destination
Recipe	Chest/Crafting Table	oak planks	CHEST / CRAFTING_TABLE	craft.item	post-handoff transform; output
Recipe	Planks/Sticks	oak planks	PLANKS / STICKS	craft.item / resource.deliver	post-handoff transform; output
Allocation	Deposit/Build	cobblestone	BUILD / DEPOSIT	build.component / resource.deliver	assigned remaining work
Allocation	Dual Build	oak planks	ROOF / WALL	build.component	assigned component

Appendix

Table S4. The eight task templates and the assignment fields that distinguish their alternatives.

Record stage	Target object	Rows
Sender proposal	multi-step SELF plus peer REQ	960
Receiver after request	downstream SELF ; empty REQ	960
Intermediate sender (single-step)	next local action after obtain	960

Appendix

Table S5. Reciprocal supervision for the sender proposal and receiver continuation in the forward interaction.

Model or condition	Rows	Stage counts	Epochs	Steps	Seeds
Multi-step / direct / surface	1,920	960 sender + 960 receiver	2	240	3
Self-plan-only	1,920	960 sender + 960 receiver	2	240	3
Matched-SFT free-form	2,400	1,200 sender + 1,200 receiver	2	300	3
Single-step	2,880	960 sender + 960 intermediate sender + 960 receiver	2	360	3
Factor backend	1,920	960 sender code + 960 receiver code	2	240	3

Appendix

Table S6. Training examples and optimization updates per checkpoint for each prediction and control protocol.

Field	Multi-step	Single-step
(a) Common configuration
Base model	Qwen3.5-0.8B ( Qwen Team, 2026 )
Optimizer	AdamW; LR $10^{-5}$ ; weight decay 0.01; cosine schedule; warm-up 0.03; grad-norm 1.0
Batch / precision	micro 4; accumulation 4; effective 16; bfloat16 on A100 40GB
Length / decoding	maximum 2,048; deterministic decoding; structured cap 256 tokens
Objective / selection	target-and-EOS causal loss; final-step checkpoint; DEV reserved for implementation checks

Appendix

Table S7. Model configuration and supervision budgets for the horizon comparison.

Pool	Changed or held out	Retained information	Counts / unit
Core requests	New instances within eight templates	Same skill ontology, task schema and public candidates	160 semantic clusters; roles and three seeds nested
Template horizon	Four whole templates absent from training	Same four families and skill ontology; task structure supplied	80 clusters, 480 evaluations per learned method
Surface/backend	Separately sampled instances, within known template families	Same shared task structure and execution path	128 quality clusters; independent 40-cluster timing samples
Workcell feedback	New instances of the four goal–mode cells	Same two goals and two modes; prescribed initial proposal	40 clusters; 20 paired mode blocks
Description wording	Lexical roots and paired distractors	Sixteen semantic choices seen in training	32 shared original phrasings; 320 role rows per condition
API request transfer	New paired blocks and another formation backend	Public task structure and four worked examples	80 pairs; 960 assignments per backend/condition

Appendix

Table S8. Evaluation design by comparison. Each row states what changes, which information remains available, and which units support inference.

Component	Decision input	Reference or implementation boundary
Forward sender	Sender-local fact and shared task structure	Model input contains the local task fact; evaluation uses the target after generation
Forward receiver	Local post-handoff context and delivered request	The expected object is built from the delivered request; raw-output consistency is checked before admission
Workcell initial requester	Private goal and public alternatives	Peer mode is absent; exact match to the prescribed initial object gates revision
Workcell revision	Local context, initial-proposal history, delivered response	History stores the canonical initial object only after equality with the actual parsed initial output
Deterministic response policy	Requested suffix and peer-local processing mode	Deterministic finite alternatives; no language-model call
Core materializer	Admitted binding and shared task structure	Instantiates task actions and dependencies from the admitted binding and shared graph

Appendix

Table S9. Information available when a component forms, validates, or evaluates work. The columns distinguish decision inputs from validation and task references.

Method	Success	Calls	Tokens		Wire B	TTC (s)
Method	Success	Calls	Input	Output	Wire B	p50	p95
Short DSL GenCoord (0.8B)	100.0%	2.0	949.1	78.6	76.3	2.27	2.62
JSON GenCoord (0.8B)	100.0%	2.0	1,154.7	267.6	303.4	7.07	7.86
Controlled free-form + structured commit (0.8B)	100.0%	2.5	1,585.2	265.8	1,061.1	7.13	7.86

Appendix

Table S10. Communication surfaces at matched task success. Quality uses 128 clusters, while calls, tokens, peer payload, and TTC use the independent 40-cluster surface timing sample.

Surface	p50 (s)	p95 (s)
Short DSL	1.14	1.66
JSON	3.53	5.94
Controlled free-form	3.27	3.75

Appendix

Table S11. Reliability and call latency for each communication surface. Reliability uses the 128-cluster quality pool and latency uses the 40-cluster timing sample.

Backend	Contrast	Estimate	95% CI
Local 0.8B	Peer-traffic reduction (%)	92.8	$[92.7,93.0]$
	Median TTC reduction (%)	68.2	$[67.9,68.7]$
	JSON output-token ratio	3.40	$[3.37,3.44]$
	Free-form output-token ratio	3.38	$[3.34,3.42]$
	JSON median-TTC ratio	3.12	$[3.08,3.16]$
	Free-form median-TTC ratio	3.15	$[3.11,3.20]$

Appendix

Table S12. Cost contrasts relative to Short DSL. Reductions compare Short DSL with the natural-language protocol; ratios divide the named protocol by Short DSL.

Family / metric	Multi-step	Single-step	$\Delta$	95% CI
Destination	100.0%	100.0%	0.0	—
Recipe	100.0%	100.0%	0.0	—
Allocation	92.5%	83.3%	9.2	—
Active branch	100.0%	81.7%	18.3	—
Overall success	98.1%	91.3%	6.9	$[2.9,10.8]$
Decisions / episode	1.98	2.91	$-0.93$	$[-0.97,-0.89]$

Appendix

Table S13. Multistep versus single-step control on 80 held-out-template clusters. Differences are multistep minus single-step, except the final rule comparison. Success differences use percentage points and decision/token differences use counts per episode, with intervals for aggregate paired effects.

Protocol	Assigned	C	H	F	G
Short DSL	768	768	768	768	768
Dialogue + structured submission	768	767	767	767	766

Appendix

Table S14. Task outcomes with the same prompted DeepSeek model. C counts formed commitments, H verified handoffs, F fulfilled commitments, and G successes on the original task. Counts use all assigned trials.

Protocol	Roles	Assigned	Formed	Fulfilled	Task success
Short DSL	Original	384	384	384	384
Short DSL	Swapped	384	384	384	384
Dialogue	Original	384	384	384	383
Dialogue	Swapped	384	383	383	383

Appendix

Table S15. Task outcomes by role orientation. Original and swapped roles each contain 384 assigned trials per protocol.

Metric	Short DSL	Dialogue	Dialogue / DSL
Logical calls	2.00	2.12	1.06
Peer traffic (B)	112.75	913.76	8.10
Input tokens	5,573	21,086	3.78
Output tokens	100.4	276.3	2.75
Estimated charge (USD)	0.000284	0.000735	2.59

Appendix

Table S16. Mean communication and model costs per assigned quality trial. Dialogue token and charge means use 767 known usage entries; other means use 768 entries. Input includes worked examples, context encoding, and interaction history, and charges price both input and output under the stated tariff.

Protocol	Task success	TTC mean	Median	p95	Calls	API charge
Short DSL	63/64	4.29	4.38	5.87	2.00	0.000257
Dialogue	64/64	6.17	5.52	10.03	2.11	0.000599

Appendix

Table S17. Formation time and estimated charges in the fresh timing sample. Each protocol has 64 assigned and formed trials. Times are seconds, and charges are USD per trial. Median averages the central observations; p95 uses the lower empirical 95th-percentile order statistic.

Condition	Assigned	C	H	F	G
Correct response, initial proposal supplied	480	419	419	419	419
Alternative response, initial proposal supplied	480	403	403	403	0
Initial proposal and revision generated	480	414	414	414	414

Appendix

Table S18. Requester revision with the prompted DeepSeek model. C, H, and F measure formation, handoff, and fulfillment; G measures success under the unchanged original task.

	Original roles		Swapped roles
Condition	Formed	Task success	Formed	Task success
Correct response	208	208	211	211
Alternative response	208	0	195	0
Full interaction	202	202	212	212

Appendix

Table S19. Feedback outcomes by role orientation. Each entry uses 240 assigned trials. The paired effect compares correct with alternative feedback over 40 mode pairs.

Condition	Assigned	Executed	$H$	$F$	$G$
Base PATH	456	456	456	456	456
GOAL with a different legal path	100	100	100	100	100
Compatible alternative request	16	16	16	16	0
Incompatible alternative request	16	0	–	–	–

Appendix

Table S20. Physical outcomes when the sender keeps its task graph private. Compatible replacements can fulfill their delivered commitment while leaving the unchanged original task unsatisfied.

Cache	Delivery	Message	Request (B)	ACK (B)	Total (B)	Cache (B)
Off	Any	PATH	605.63	135.00	740.63	0.00
Off	Any	SLICE	670.30	135.00	805.30	0.00
Off	Any	GOAL	526.89	135.00	661.89	0.00
On	Cold	PATH	605.63	135.00	740.63	348.88
On	Cold	SLICE	670.30	135.00	805.30	401.21
On	Cold	GOAL	526.89	135.00	661.89	273.31

Appendix

Table S21. Transmitted bytes for paths, slices, and goals under first, cached, and changed delivery. Requests include content or a cache reference, every acknowledgment is 135 B, and receiver cache is stored content rather than transmission. With caching off, Any denotes identical costs across the three delivery conditions.

Cache	Delivery	PATH	SLICE	GOAL
Off	Cold	65	66	2010
Off	Warm	60	61	2026
Off	Changed	60	62	1983
On	Cold	58	60	1984
On	Warm	50	52	1981
On	Changed	58	60	1989

Appendix

Table S22. Delivery-processing CPU time in microseconds (

\mu

s), including serialization, decoding, realization, checks, and acknowledgment. Entries are medians of setting-level medians after pairing roles and versions, using five technical repeats and excluding model calls and world execution.

Checkpoint seed	Calls	Strictly formed	Formation (%)
Seed 1	384	223	58.1
Seed 2	384	145	37.8
Seed 3	384	128	33.3
All	1,152	496	43.1

Appendix

Table S23. Complete sender objects generated by each unchanged Direct checkpoint under private task structure.

Metric	Factor + rule	Direct DSL
Held-out success	768/768	768/768
Input tokens	980.2	948.9
Output tokens	19.4	79.1
Protocol-specific peer payload	76.6 B	76.6 B
TTC p50	0.82 s	2.45 s
TTC p95	1.00 s	2.72 s

Appendix

Table S24. Direct and Factor formation at matched quality. Success uses 128 clusters, with two roles and three checkpoints, while generation cost and TTC use a separate 40-cluster timing sample.

Method	Task success
Complete-case table	0/60
Explicit-factor composer	60/60
Direct Short DSL	60/60

Appendix

Table S25. Complete-case lookup versus field-wise composition on 60 unseen assignment cross-products.

Sender source	Learned	Deterministic path	Directory lookup
API (960)	952	959	959
Local (960)	958	958	958
All (1,920)	1,910	1,917	1,917

Appendix

Table S26. Learned, deterministic, and directory-based receivers applied to the same saved sender outputs.

Implementation	Median (ms)	Mean (ms)	p95 (ms)	Maximum (ms)
Learned	856.11	880.84	972.99	17482.75
Deterministic path	0.63	0.63	0.65	1.21
Directory lookup	0.36	0.36	0.38	0.73

Appendix

Table S27. Receiver processing latency on the common local sample. Each method has 960 operations over 16 world blocks, including all technical repetitions and the long learned-receiver observation.

Seed	Learned	Deterministic path	Directory lookup
Seed 1	916.77	0.63	0.36
Seed 2	872.80	0.63	0.36
Seed 3	852.94	0.62	0.36

Appendix

Table S28. Mean receiver processing time in milliseconds by local checkpoint, with 320 operations per method and checkpoint.

Configuration	Sender formation	Receiver
Direct / learned	Path generation	Learned path generation
Direct / deterministic	Same Direct sender output	Deterministic interpretation
Factor / learned	Short-code prediction and composition	Learned code and composition
Factor / deterministic	Same Factor sender output	Deterministic interpretation
TF–IDF retrieval	Nearest TRAIN description, then composition	Deterministic interpretation
Supervised lexical classifier	TRAIN-fitted logistic regression, then composition	Deterministic interpretation

Appendix

Table S29. Assignment and receiver implementations for natural descriptions. Generation produces a path or code, while an embedding call encodes the query description. The privileged reference supplies the correct assignment for quality evaluation.

Figure S3. From natural descriptions to world outcomes. Panel (a) shows mutually exclusive results over 10,240 assigned trials, normalizing each configuration by its own denominator and reporting task successes and losses after formation at right. Panel (b) compares success and mean TTC on the same fresh timing sample, with arrows and paired 95% intervals for receiver replacement over 16 sources. The lexical mean includes the 17.3 s observation, while its median is 0.06 s. Panel (c) reports success for 16 choices, each derived from two phrasings: hatching marks no formed commitment and red ticks mark formation below 100%. The privileged reference completes every choice and has no timing arm. Three panels. Panel a shows one horizontal bar per configuration divided into task success, fulfilled obligations with failed tasks, completion failures, handoff failures, and unformed commitments; embedding retrieval reaches 90.9 percent task success, Factor about 79 percent, Direct about 65 percent, and the lexical selectors about 58 to 61 percent, with at most five trials lost after formation per configuration. Panel b plots task success against mean time to commitment on the fresh timing sample; arrows from learned to deterministic receivers shorten time to commitment by 785 milliseconds for Direct and 343 milliseconds for Factor at the same observed task success. Panel c is a heat map of task success for seven configurations across sixteen semantic choices, marking cells where no commitment formed.

Configuration	Assigned	$C$	$H$	$F$	$G$	$G$ (%)
Paraphrases
Direct / learned	960	898	898	898	632	65.8
Direct / deterministic	960	898	898	898	632	65.8
Factor / learned	960	900	899	899	758	79.0
Factor / deterministic	960	900	900	899	759	79.1
TF–IDF retrieval	320	320	319	319	189	59.1

Appendix

Table S30. Physical outcomes under each description condition. All assigned trials contribute to C (formation), H (verified handoff), F (declared fulfillment), and G (original-task success), with task success also shown as a percentage.

Paraphrases
Contrast	Difference	Stratified 95% CI	Unstratified 95% CI
Factor: deterministic $-$ learned	$+0.1$	$[0.0,0.2]$	$[0.0,0.3]$
Direct: deterministic $-$ learned	$0.0$	$[0.0,0.0]$	$[0.0,0.0]$
Factor deterministic $-$ Direct learned	$+13.2$	$[10.4,16.3]$	$[-3.0,29.7]$
Factor deterministic $-$ TF–IDF	$+20.0$	$[12.0,28.0]$	$[2.5,38.1]$
Factor deterministic $-$ supervised lexical	$+19.7$	$[12.0,27.4]$	$[2.2,38.0]$

Appendix

Table S31. Paired differences in task success, in percentage points, for each selector and receiver configuration. Stratified and unstratified intervals use the same source pairing.

Paraphrases
Contrast	Metric	Difference	Stratified 95% CI	Unstratified 95% CI
Factor: deterministic $-$ learned	$C$	$0.0$	$[0.0,0.0]$	$[0.0,0.0]$
Factor: deterministic $-$ learned	$F$	$0.0$	$[-0.2,0.2]$	$[-0.3,0.3]$
Direct: deterministic $-$ learned	$C$	$0.0$	$[0.0,0.0]$	$[0.0,0.0]$
Direct: deterministic $-$ learned	$F$	$0.0$	$[0.0,0.0]$	$[0.0,0.0]$
Factor deterministic $-$ Direct learned	$C$	$+0.2$	$[-1.9,2.3]$	$[-8.4,6.9]$

Appendix

Table S32. Paired changes in commitment formation and declared fulfillment, in percentage points. Both use the same source resampling as the task-success comparison.

Configuration	$n$	$G$	TTC mean	TTC median	TTC p95	Episode mean	Episode median	$V(30)$	$V(60)$
Direct / learned	192	125	2.35	2.29	2.56	24.09	24.98	64.6	65.1
Direct / deterministic	192	125	1.57	1.55	1.72	23.29	24.18	65.1	65.1
Factor / learned	192	154	0.78	0.78	0.90	22.31	23.55	80.2	80.2
Factor / deterministic	192	154	0.44	0.43	0.51	21.97	23.14	80.2	80.2
TF–IDF retrieval	64	42	0.07	0.06	0.11	22.94	22.66	65.6	65.6
Supervised lexical	64	42	0.34	0.06	0.13	23.21	22.68	64.1	65.6

Appendix

Table S33. Latency and verified deadline success on the fresh sample. Times are seconds over all assigned trials, including rejection time. TTC ends at commitment, episode time includes world execution, and

V(30)

V(60)

give percentages completing the task by 30/60 s.

Contrast	$\Delta$ TTC (ms)	95% CI	$\Delta$ episode (ms)	95% CI
Factor: deterministic $-$ learned	$-343$	$[-374,-292]$	$-335$	$[-378,-279]$
Direct: deterministic $-$ learned	$-785$	$[-936,-681]$	$-800$	$[-956,-680]$
Factor deterministic $-$ Direct learned	$-1,918$	$[-2070,-1812]$	$-2,114$	$[-5679,540]$
Factor deterministic $-$ TF–IDF	$+371$	$[359,383]$	$-963$	$[-3891,634]$
Factor deterministic $-$ supervised lexical	$+99$	$[-448,379]$	$-1,239$	$[-4291,543]$
Factor deterministic $-$ embedding	$+319$	$[301,335]$	$-1,016$	$[-3926,529]$

Appendix

Table S34. Paired changes in TTC and episode time, in milliseconds. A negative value denotes shorter time for the first configuration, with every assigned repeat included.

Configuration	Gen. calls	Input tokens	Output tokens	REQ bytes	ACK bytes
Direct / learned	1.95	943.1	76.9	71.5	64.5
Direct / deterministic	1.00	455.0	53.3	71.5	64.5
Factor / learned	1.94	968.1	18.5	72.4	63.8
Factor / deterministic	1.00	457.0	9.6	72.4	63.8
TF–IDF retrieval	0.00	0.0	0.0	75.2	68.0
Supervised lexical classifier	0.00	0.0	0.0	75.2	68.0

Appendix

Table S35. Mean work per assigned timed trial. Generation tokens and embedding-query tokens are distinguished, and peer traffic comprises the transmitted request and acknowledgment.

Formation method	Recombination: task success	Paraphrases: binding accuracy	With distractors: binding accuracy
Direct Short DSL (0.8B)	100.0%	65.8%	64.8%
Learned factor + rule (0.8B)	100.0%	79.1%	80.1%
Exact-code parser rule	100.0%	0.0%	0.0%
TF–IDF nearest-neighbour rule	100.0%	59.4%	62.5%
Supervised TF–IDF + logistic regression	—	59.4%	57.5%
Complete surface table	0.0%	0.0%	0.0%

Appendix

Table S36. Assignment from descriptions under three language conditions. Syntactic recombination reports Minecraft task success, while paraphrase and distractor columns report planning accuracy. Each uses 320 role instances and three checkpoints for learned methods, with the latter two conditions derived from 32 phrasings across 16 choices.

Contrast	Difference	Stratified 95% CI	Unstratified 95% CI
Paraphrases: Factor $-$ retrieval	19.7	$[12.0,27.3]$	$[2.3,37.8]$
With distractors: Factor $-$ retrieval	17.6	$[11.0,24.0]$	$[0.4,35.2]$
Paraphrases: Direct $-$ retrieval	6.5	$[-0.4,13.3]$	$[-7.5,21.8]$
With distractors: Direct $-$ retrieval	2.3	$[-3.5,8.1]$	$[-12.1,17.0]$

Appendix

Table S37. Paired planning-accuracy differences in percentage points. Source-stratified intervals hold the mixture of 16 choices fixed, while unstratified intervals allow that mixture to vary. Retrieval denotes TF–IDF nearest-neighbor lookup.

Strategy	Assigned	$C=H=F$	$G$	$G$ (%)	$F-G$	Factor (%)
Embedding retrieval	1,280	1,280	1,052	82.2	228	0.0
Factor	3,840	3,840	3,684	95.9	156	100.0
Margin-guided upgrade	3,840	3,840	3,395	88.4	445	15.2
Random upgrade	3,840	3,840	3,268	85.1	572	15.2
Score-guided upgrade	3,840	3,840	3,280	85.4	560	15.2
Margin-based abstention	1,280	1,086	944	73.8	142	0.0

Appendix

Table S38. Prospective task outcomes for all six strategies. C, H, and F coincide within each row, while

F-G

counts fulfilled commitments that select the wrong work. Factor use is the percentage of assigned trials that invoke its generator.

Contrast	Metric	Effect	Level (%)	Stratified CI	Unstratified CI
Margin $-$ maximum similarity	AUROC	+0.176	97.5	$[0.109,0.245]$	$[0.107,0.247]$
Upgrade $-$ retrieval	$G$ (pp)	+6.2	97.5	$[4.4,8.2]$	$[4.2,8.4]$
Upgrade $-$ random	$G$ (pp)	+3.3	95	$[1.5,5.2]$	$[1.5,5.2]$
Upgrade $-$ score	$G$ (pp)	+3.0	95	$[1.1,4.9]$	$[1.1,4.9]$
Upgrade $-$ Factor	$G$ (pp)	-7.5	95	$[-9.9,-5.2]$	$[-10.0,-5.1]$

Appendix

Table S39. Paired prospective effects over 640 descriptions. Confidence levels are shown per row, and both interval schemes average repeated roles and checkpoints within description.

Strategy	$n$	$C$	$H$	$F$	$G$	TTC	Episode	Factor calls
Embedding retrieval	128	128	128	128	116	91	22.98	0.000
Margin-guided upgrade	384	384	384	384	366	144	23.01	0.125
Factor	384	384	384	383	373	448	23.30	1.000
Margin-based abstention	128	112	112	112	106	93	20.20	0.000

Appendix

Table S40. Formation time, physical outcomes, and generator use on the fresh timing sample. TTC is mean milliseconds, episode time is mean seconds, and Factor calls are per assigned trial.

Contrast	Metric	Difference	95% CI
Margin upgrade minus retrieval	Success (pp)	+4.7	$[0.0,9.4]$
Margin upgrade minus retrieval	TTC (ms)	+53	$[34,71]$
Margin upgrade minus retrieval	Episode (ms)	+29	$[-9,64]$
Margin upgrade minus retrieval	Factor calls	+0.125	$[0.078,0.172]$
Margin upgrade minus Factor	Success (pp)	-1.8	$[-6.5,2.3]$
Margin upgrade minus Factor	TTC (ms)	-304	$[-355,-261]$

Appendix

Table S41. Paired effects on the fresh sample with 95% intervals over 64 descriptions. Differences are first strategy minus second, so negative timing effects denote savings.

Choice	Retrieval	Factor	Margin	Random	Score	Abstain
Active order: storage	100.0	98.3	100.0	100.0	100.0	100.0
Active order: work	65.0	99.2	75.0	65.0	70.0	60.0
Build site A	92.5	97.5	95.0	95.0	95.0	82.5
Build site B	82.5	100.0	95.0	82.5	90.0	40.0
Chest A	100.0	100.0	100.0	100.0	100.0	85.0
Chest B	40.0	100.0	87.5	75.0	72.5	10.0

Appendix

Table S42. Task success by semantic choice, in percent. The three upgrade strategies have matched quotas, and abstention counts unsubmitted trials as unsuccessful.

Checkpoint	Factor	Margin	Random	Score
Checkpoint 1	1,216 (95.0)	1,133 (88.5)	1,088 (85.0)	1,094 (85.5)
Checkpoint 2	1,207 (94.3)	1,126 (88.0)	1,086 (84.8)	1,090 (85.2)
Checkpoint 3	1,261 (98.5)	1,136 (88.8)	1,094 (85.5)	1,096 (85.6)

Appendix

Table S43. Task success for the existing Factor checkpoints, with counts out of 1,280 followed by percentages. The retrieval reference has 1,052 successes for every checkpoint comparison.

Checkpoint	Routing group	Trials	Corrected	Changed to wrong	Both wrong	Both correct
1	Retrieval kept	1086	115	32	27	912
1	Upgraded	194	82	1	4	107
2	Retrieval kept	1086	117	36	25	908
2	Upgraded	194	80	6	6	102
3	Retrieval kept	1086	138	13	4	931
3	Upgraded	194	84	0	2	108

Appendix

Table S44. Assignment changes caused by replacing retrieval with Factor. Corrected and introduced errors use matched descriptions and roles, with full Factor providing the comparison on descriptions retained by retrieval.

Selector	Correct/assigned	Accuracy (%)	Time (ms)	API USD
Embedding retrieval	1,052/1,280	82.19	92	0.00000
Factor	3,684/3,840	95.94	412	0.00000
DeepSeek	3,839/3,840	99.97	1402	0.00009
Retrieval + DeepSeek	3,414/3,840	88.91	350	0.00001
Random upgrade (expectation)	–	84.88	–	–

Appendix

Table S45. Assignment accuracy and measured decision cost. Correct assignments are determined before execution on 640 descriptions, with two roles and three repetitions for generative selectors. Timing uses 128 decisions per strategy. API charges are usage estimates.

Figure S4. Local content-substitution checks. (a) Correct and alternative feedback select different workcell assignments, while local planning and centralized information provide information references. (b) Correct, absent, and alternative requests test whether the receiver selects the branch communicated by the sender. Each axis measures success against the original task. The two original local content diagnostic panels. Feedback gives 100 percent success with the correct response or centralized information, zero with counterfactual feedback, and 50 with local planning. Requests give 100 percent with the correct message or a binding rule, zero with the legal alternative, and 50 without a request.

Condition	Assigned	C	H	T	F	G
Clean	256	1	1	1	1	1
Irrelevant change	256	1	1	1	1	1
Malformed proposal	256	0	–	–	–	0
Incompatible handoff	256	0	–	–	–	0
Insufficient handoff	256	1	0	–	0	0
Wrong transfer item	256	1	0	–	0	0

Appendix

Table S46. Stage checks under controlled faults. Every row describes 256 trials with the shown outcome, where a dash denotes a stage that is not reached. T is declared terminal completion.

Condition	Episodes	Success	Calls	Request (B)	$\Delta$ vs. true	95% CI
True learned request	960	100.0%	2.0	304.4	–	–
Request removed (self-plan-only)	960	50.0%	2.0	0.0	$+50.0$	$[42.5,57.5]$
Same-template executable alternative	960	0.0%	2.0	304.4	$+100.0$	$[100,100]$
Deterministic correct binding	320	100.0%	0.0	304.4	$0.0$	–

Appendix

Table S47. Request substitutions and a deterministic assignment reference over 160 clusters. Learned methods use three checkpoints and 960 episodes, while the rule uses 320 role views. Differences are true request minus each condition, in percentage points. Bytes count canonical request objects rather than Short DSL surface strings.

Model	Visible information	Rows	Epochs	Updates	Train-token occurrences	Seeds
Distributed requester	320 initial local + 320 after-response local	640	2	80	868,720	3
Centralized full information	merged current local views	320	4	80	970,080	3

Appendix

Table S48. Workcell feedback and centralized training budgets. Token occurrences include prompt, target, and end markers across the training epochs.

Condition	Seeds	Epis.	Success	95% CI	Strict	Calls	Measured boundary component
Requester-local (A-only)	3	240	50.0%	$[35,65]$	20/40	1.0	0 B cross-boundary payload
Correct bounded feedback	3	240	100.0%	$[100,100]$	40/40	2.0	response: mean 103 B
Counterfactual feedback	3	240	0.0%	$[0,0]$	0/40	2.0	injected response: mean 103 B
Centralized full information	3	240	100.0%	$[100,100]$	40/40	1.0	state aggregation: mean 956 B

Appendix

Table S49. Workcell task success and information components over 40 clusters forming 20 mode pairs. Intervals use the cluster bootstrap, and strict success requires all six role–checkpoint repetitions to succeed. The final column measures the stated message component.

Field	Matches injected	Matches true world	Observed behavior
Craft actor	240/240	0/240	rewritten
Handoff item	240/240	0/240	rewritten
Handoff count	240/240	0/240	rewritten
Peer suffix	240/240	0/240	rewritten
Goal item	240/240	240/240	invariant
Goal count	240/240	240/240	invariant

Appendix

Table S50. Assignment fields after counterfactual feedback, separating the altered division of work from the unchanged requested product.

Backend	Condition	Formed (%)	Success (%)	S	R	C	W
DeepSeek	Correct request	949 (98.9)	949 (98.9)	0	9	2	0
DeepSeek	Absent request	795 (82.8)	385 (40.1)	0	165	0	410
DeepSeek	Alternative request	957 (99.7)	0 (0.0)	0	3	0	957
DeepSeek	Full protocol	952 (99.2)	952 (99.2)	1	7	0	0
Local	Correct request	960 (100.0)	960 (100.0)	0	0	0	0
Local	Absent request	960 (100.0)	480 (50.0)	0	0	0	480

Appendix

Table S51. Prompted and local request-transfer outcomes over 960 trials per condition. Formation and success are counts with percentages, while failure columns partition sender formation, receiver formation, request conflict, and formed commitments failing the original task.

Backend	Contrast	Effect (pp)	95% CI
DeepSeek	Correct minus legal alternative	$+98.9$	$[98.1,99.5]$
DeepSeek	Correct minus absent	$+58.8$	$[56.8,60.8]$
Local	Correct minus legal alternative	$+100.0$	$[100.0,100.0]$
Local	Correct minus absent	$+50.0$	$[50.0,50.0]$

Appendix

Table S52. Paired task-success differences in percentage points, with 95% intervals from resampling 80 paired world blocks within template.

Figure S5. Request and feedback outcomes by evaluation unit. Panels (a,b) show the local and prompted backends on 80 paired world blocks, and panel (c) shows 40 workcell clusters by processing mode. Each cell averages six role–repetition or role–checkpoint trials, with pooled rates beside the rows. Panel (d) relates the choice made without a request to the true assignment, while None denotes the absence of a formed plan. The outcomes provide a unit-level check of the paired-information construction in Section S12.1 . Barcode heat maps with one cell per world block or cluster and marginal bars of pooled task success. For both local checkpoints and the DeepSeek API, correct requests and end-to-end runs succeed in nearly every block and alternative requests fail in every block; absent requests succeed mainly on the first binding. For feedback, correct and centralized conditions succeed in all clusters, alternative feedback fails in all clusters, and local planning succeeds only for the B-crafts mode. Two confusion tables show that without the request both backends mostly select the first binding.

Figure S6. Task structure and presentation order. Panel (a) crosses stage labels with original or canonical node order and reports parsing, assignment, and complete-plan counts out of 576. Panel (b) gives paired factorial effects with 95% cluster intervals. Panel (c) compares the Full, Skeleton, and Node-only projections over 32 clusters per motif, using two roles and three checkpoints for learned methods and a seedless symbolic rule. Blue and red nodes indicate sender and receiver work. Panel a is a two-by-two grid of schematic node sequences crossing stage labels present or removed with original or canonical order, each with bars for parse validity, binding correctness, and full-plan correctness out of 576. Panel b is a forest plot of order, stage-label, and interaction effects for the three outcomes; the order effect on full-plan correctness is about thirty points. Panel c is a heat map of task success for three methods across Full, Skeleton, and Node-only views for three path motifs.

Multi-Agent Coordination

Multi-agent coordination studies how autonomous agents align actions, share information, and divide work to achieve individual or collective goals.

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Collective Counterfactual Planning: Coordination, Consent, and Verification under Representational Constraints
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Aug 18, 2026J/K move · Enter open · S save

Collective Counterfactual Planning: Coordination, Consent, and Verification under Representational Constraints

Chainarong Amornbunchornvej

National Electronics and Computer Technology Center (NECTEC), NSTDA Pathum Thani, Thailand

Multi-Agent Coordination Multi-Agent Planning

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Groups routinely complete projects that no single member can plan, execute, or verify alone. We propose a formal model of this phenomenon, Collective Counterfactual Planning (CCP), in which the binding limitation on each agent is neither capability, knowledge, nor observability, but representational geometry: each agent perceives the state, conceives moves, consents to actions, and certifies goal requirements only through a projection onto an agent-specific subspace of a common task space. Four gates jointly determine whether a team can reach a conjunctive goal and legitimately recognize that it has done so: the exogenous implementation coalitions required to perform each action, together with three representational gates -- conception, consent, and task-relative verification qualification. We define the Collective Counterfactual Solvability (CCS) problem, separating geometric feasibility, executable attainment, and validated completion. The results expose a positive-negative duality. Iterated cross-agent relay can unlock a solution that no one-shot pooling of individual plans contains, but any goal requirement depending essentially on the subspace dark to the entire team is unverifiable and therefore not validly completable, even when the trajectory accidentally attains it. Memoryless and audited consent further constrain different objects -- action directions versus cumulative trajectory states -- and neither dominates the other. A four-step exhaustive horizon-bounded solvability scheme is sound and complete under exact representation of the relay closure; restricted implementations remain sound on returned plans but need not be complete. The model gives one geometry for sequential mutual enabling, competent execution of steps whose purpose is invisible to the executor, forced sub-teaming at expertise boundaries, and completion that cannot be validly declared.

Abstract

Multi-Agent Coordination

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