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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

French

Les systèmes de registre multi-agent peuvent différer sous des agents identiques et des budgets de communication lorsqu'ils prennent en charge différentes formes d'interaction conjointe. Nous introduisons les fiches topologiques d'irréductibilité primaire (TSI), un cadre théorique de tâche séparant les obstacles à la compatibilité au niveau de la tâche de la capacité du système. Une tâche est représentée par un complexe d'interaction $K$ et une capacité de coupe $\mathcal{F}$; nonzero $H^n(K;\mathcal{F})$ définit une signature structurelle, tandis que les capacités d'essai d'intervention enregistrées. Pour les données de co-cycle, la réalisation de la cochain est résolue exactement

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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