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Topological Sheaves of Irreducibility: A Theoretical Framework for Multi-Agent Task Obstructions

Steven Motta, Davide Mambelli, Alessandro Rozza · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.26777 · cited by 0 works (OpenAlex) · open-access PDF

Japanese

複数のエージェントのレジストリシステムは、同一のエージェントと通信予算の下で異なりますが、相互作用の異なる形態をサポートします。私たちは、タスクレベル互換性の障害をシステム能力から分離するシェイフ理論的枠組みであるCondological Sheaves of Primary Irreducibility (TSI)を導入します。タスクレベル互換性の障害をシステム能力から区別するタスクレベルレジストリシステムです。タスクは、相互作用の複雑なK$と容量シェイフ $\mathcal{F}$によって表されます。 サインが sheaf-relativeであることを示す例と、タスクレベル1の減少は、同一のサイクル空間と観察要件を定義します。

Machine translation (Meta M2M100); the English original follows.

Abstract

Multi-agent systems can differ under identical agents and communication budgets when they support different forms of joint interaction. We introduce Topological Sheaves of Irreducibility (TSI), a sheaf-theoretic framework separating task-level compatibility obstructions from system capability. A task is represented by an interaction complex $K$ and a capacity sheaf $\mathcal{F}$; nonzero $H^n(K;\mathcal{F})$ defines a structural signature, while registered interventions test capability. For cocycle data, cochain realization is solvable exactly when the represented cohomology class vanishes. Examples show that the signature is sheaf-relative, and the degree-1 reduction yields cycle-space witnesses and observation requirements. We evaluate two finite one-round models. Across 15,625 degree-1 inputs, an all-agent additive primitive solves decision and repair exactly; the optimal pairwise system reaches 88.80% on decision, one input above the 88.79% majority-class baseline, and 26.63% on repair. On a degree-2 tetrahedral-boundary task over 531,441 inputs, the same-2-skeleton system retains all four three-agent primitives yet matches the pairwise-only optimum throughout the registered separation window, whereas adding the class-filling four-agent primitive makes both tasks exact. Status-preserving robustness checks retain exact performance but are not cost-matched to the class-restoring ablation. Conditional on the registered primitives and cost models, TSI provides task-relative, falsifiable tests of higher-order interaction.

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Topics: Mathematics, Irreducibility, Pairwise comparison, Robustness (evolution), Topology (electrical circuits), Affine transformation

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