The Big Picture
A team can complete goals that no single member could plan for by sequentially enabling each other, but if every member is blind to some task dimension the team can never validly certify completion.
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The Evidence
Representational limits — what each agent can mentally see or describe — create four distinct gates on collective action: who can physically implement a move, whether someone can conceive it, whether required partners will consent, and whether anyone can verify completion. Small, otherwise-acceptable moves can enable other agents to see new moves (a relay effect), letting the team reach objectives unreachable by one-shot pooling. However, any requirement that varies along a direction invisible to the whole team cannot be certified, so accidental physical achievement is not the same as validated completion. Two different consent rules (memoryless versus audited) are neither strictly stronger nor weaker: they allow different solutions and can force sub-teams or specific orderings. This also informs approach to planning: a Planning Pattern can guide how steps are composed and validated.
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Data Highlights
1Interpretability example: cos^2(20°) ≈ 0.883 (88.3%) exceeded the consent threshold 0.8, allowing a nearly-horizontal move to be accepted.
2A small tilt produced a vertical displacement tan(20°) ≈ 0.364, which made a continuation visible to the other agent and completed the goal.
3Audited consent example: an audited interpretability ≈ 0.083 cleared a low threshold 0.05 while some single-step directions had interpretability 0 and would be rejected memorylessly.
What This Means
Engineers building multi-agent teams and orchestration systems should care because representational overlap and ordering of actions can create or block solutions even when physical capabilities exist. Technical leaders and evaluators designing agent-to-agent trust, delegation, or verification policies can use these geometric checks to decide who must be involved and which requirements need extra instrumentation or inspectors. For practical architectures, consider the Orchestrator-Worker Pattern to structure responsibilities and data flow Orchestrator-Worker Pattern.
Key Figures
Fig 3: Figure 3: A two-step counterfactual relay in E1. Agent 1 changes the state so that agent 2 can conceive a continuation that was unavailable at the start. In agent 2’s projection the whole plan is the closed loop 0 → tan α → 0 0\to\tan\alpha\to 0 , even though the group makes goal-relevant progress along a dimension dark to agent 2. The same instance is memoryless-solvable but audited-unsolvable (Theorem 2 , Proposition 3 , Theorem 4 ).
Fig 4: Figure 4: Success rates across structured random instances. Left: greater representational coverage increases attainment and validated completion. Right: audited consent helps at lenient thresholds but hurts at strict thresholds, illustrating its incomparability with memoryless consent.
Fig 5: Figure 5: Shared representation changes who can act together. As the common representational core grows, successful plans use larger coalitions and more whole-team steps; progress outside that core is carried by smaller subteams. All instances remain solvable.
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Learn MoreYes, But...
The model assumes honest, non-strategic agents and public knowledge of tokens and bases; it does not address deception, conflicting goals, or adversarial behavior. Solvability checks are exhaustive only under a finite horizon and exact relay closure; practical restricted searches are sound (any returned plan is valid) but may miss existing solutions. The framework uses a linear vector representation of concepts (subspaces and projections), so results depend on that representational choice and may not directly transfer to non-linear or learned internal representations. The verification component can be informed by a Capability Attestation Pattern to ensure robust attestations and audits Capability Attestation Pattern.
Methodology & More
Model and approach: Agents are modeled by the subspace of a shared task space they can represent — intuitively, the directions of the task they can perceive and reason about. Four gates determine collective progress: (1) implementation (who must physically act), (2) conception (an agent only proposes steps that look like progress in its own view), (3) consent (required participants accept only sufficiently interpretable steps), and (4) verification (only agents who can see all distinctions relevant to a requirement can certify it). The decision problem separates geometric feasibility (can moves add up at all?), executable attainment (can a relay of conceivabilities and consents produce the goal?), and validated completion (can the team legitimately declare the goal done?). The approach can be framed using the A2A Protocol Pattern to ensure reliable agent-to-agent communication and decision making A2A Protocol Pattern. Key findings and implications: Sequential relay matters — one agent’s small, interpretable move can change another agent’s view so a previously inconceivable continuation becomes available. Concrete examples show how a nearly-horizontal move (interpretability ≈ 88%) created a vertical slip (≈ 0.364) that let the vertical agent act and finish the task. But if a requirement depends on a direction that no agent represents (a dark subspace), no verifier exists and certification is impossible regardless of the trajectory. Two consent modes were studied: memoryless consent (vetoes directions outright) and audited consent (looks at cumulative trajectory); they are incomparable and can force specific orderings or sub-teaming. An exhaustive four-step scheme (cover preflight, relay closure, ratification, terminal inspection) is sound and complete when the relay closure is exactly represented; practical restricted searches remain sound but can be incomplete. Practically, the results tell system designers when to add overlapping representations, which agents must be able to inspect a requirement, and when institutional verifiers are necessary to guarantee valid completion. For architectural alignment with defense-in-depth concepts, consider the Defense in Depth Pattern to structure layers of checks and fallbacks Defense in Depth Pattern.
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Credibility Assessment:
Single author with modest h-index (8) at a national research center (NECTEC); arXiv preprint and no citations — some institutional signal but not strong enough for higher rating.