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

Steven Motta; Davide Mambelli; Alessandro Rozza

cs.LGOct 19, 2026

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

Authors: Steven Motta, Davide Mambelli, Alessandro Rozza

Organizations: distributed cognition

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.

Figures & tables

Panel of Figure 1 — Figure 1: Class-filling in degrees 1 and 2, with the constant sheaf $\underline{\mathbb{R}}$ . ( a ) Three agents with pairwise channels only span the boundary of a triangle and carry a degree-1 class; ( b ) the three-agent joint operation fills the 2-simplex and the class vanishes. ( c ) The tetrahedral boundary retains all four three-agent operations, yet carries a degree-2 class; ( d ) only the four-agent operation fills the 3-simplex and removes it. Dashed segments are the edges hidden behind the body. In each row the communication graph is identical on both sides: the difference is the top-dimensional joint operation, not the number of pairwise channels. Panel ( c ) is the configuration of the system $S_{\mathrm{face}}$ in Section 9.2 , whose exhaustive optimum coincides with pairwise-only communication throughout the registered separation window.

Framework	Primary object	Relation to TSI
Partial information decomposition [ 35 ]	Joint information unavailable from marginals	TSI represents interaction geometry and compatibility constraints; it assumes no information-theoretic decomposition. A direct empirical comparison is whether PID synergy and $\beta^{n}$ dissociate on tasks with matched information but different interaction structure.
Cooperative game theory [ 31 ]	Coalition value function	TSI requires no transferable utility and compares no coalition values; it identifies sheaf-relative obstructions. A direct empirical comparison is whether coalition superadditivity and TSI signatures dissociate across matched task families.
Integrated information [ 33 , 23 ]	Axiomatic integration measure	Both address irreducibility with different primitives; no equivalence is asserted. Shared benchmark tasks could support direct comparison where the corresponding integration measure is well-defined.
Sheaf-theoretic contextuality [ 1 , 2 ]	Compatibility of local measurement sections with a global section	Shares the local-to-global obstruction pattern, but TSI uses cellular interaction complexes and task-specific capacity sheaves and evaluates system capability separately through registered interventions.
Sheaf neural networks [ 8 , 17 ]	Learnable sheaves and sheaf Laplacians	One way to learn capacity sheaves from data (Section 8.2 , Recipe B).
Combinatorial topology of distributed computing [ 20 ]	Task solvability via subdivisions	Tools for turning Proposition 5.4 into model-specific round and bit lower bounds.

Table 1: Conceptual relations. Each row is an empirical comparison target, not an identification.

System	$\boldsymbol{\mathsf{T}_{1}}$	$\boldsymbol{\mathsf{T}_{3}}$	Control
$S_{\mathrm{fill}}$ (atomic $g_{A}$ )	$1.0000$	$1.0000$	$1.0000$
$S_{\mathrm{pair}}$ (same 1-skeleton)	$0.8880$	$0.2663$	$1.0000$
$S_{\mathrm{ring}}$ (sanity check)	$0.8879$	$0.1687$	$1.0000$
Majority-class baseline	$0.8879$	—	—

Table 2: Exhaustive evaluation for

m=6

B=2

, and

\beta=5

\mathsf{T}_{1}

is evaluated on all

15{,}625

inputs;

\mathsf{T}_{3}

on the

13{,}874

unrealizable inputs.

$\boldsymbol{\beta}$	$\boldsymbol{S_{\mathrm{fill}}}$	$\boldsymbol{S_{\mathrm{face}}}$	$\boldsymbol{S_{\mathrm{pair}}}$	Majority baseline $\mathsf{T}^{(2)}_{1}$
$5$	$1.0000/1.0000$	$0.8884/0.4666$	$0.8884/0.4666$	$0.8612$
$6$	$1.0000/1.0000$	$0.9295/0.6299$	$0.9295/0.6299$	$0.8612$
$7$	$1.0000/1.0000$	$0.9453/0.7193$	$0.9453/0.7193$	$0.8612$
$8$	$1.0000/1.0000$	$0.9681/0.8401$	$0.9681/0.8401$	$0.8612$

Table 3: Exhaustive evaluation across the complete registered separation window for the degree-2 task model

(\partial\Delta^{3},\underline{\mathbb{R}})

with

B=1

. Each system entry reports

(\mathsf{T}^{(2)}_{1},\mathsf{T}^{(2)}_{3})

;

\mathsf{T}^{(2)}_{1}

is evaluated on all

531{,}441

inputs and

\mathsf{T}^{(2)}_{3}

on the

457{,}652

unrealizable inputs. The trivial-cohomology control is solved exactly by every system at every reported budget.

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Complex tasks inherently couple workflow structure, agent responsibility, collaboration, and memory access: task regions delimit responsibility and tool scope, cross-region dependencies give rise to handoffs, and ownership boundaries delimit private and selectively shared memory. Existing methods can jointly optimize agent and communication structures, yet such optimization does not by itself require responsibility, handoff, and memory boundaries to remain consistent with task dependencies. We term this requirement topological coherence. We introduce TOCOMAS, a Topology-Coherent Multi-Agent System. TOCOMAS grounds a task graph in tool interfaces, organizes compatible task nodes into reusable responsibility domains, and derives dependency-induced and profile-conditioned collaboration together with boundary-regulated memory visibility. During online self-evolution, TOCOMAS proposes coupled changes to agent, collaboration, and memory policies, retaining for subsequent tasks only candidates that satisfy structural constraints and improve evaluated reward. Across BBEH, WorkBench, SWE-Bench-Verified, and CoMemBench, TOCOMAS improves task success over baselines across backbones. CoMemBench also shows gains over the self-evolving baseline in verified progress, handoffs, and memory isolation.

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