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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