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A Sheaf Framework for Strategic Multi-Agent Systems: From Consensus to Nash Equilibria
Manuel Hernández, Eduardo Sánchez-Soto
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
Last extracted: 7/9/2026, 12:01:46 AM
Summary
The paper proposes a unified categorical framework for strategic multi-agent systems using sheaf theory and topos semantics. It introduces a game sheaf where global sections correspond to Nash equilibria, and sheaf cohomology identifies strategic inconsistencies. The model integrates event calculus and reward structures with geometric consensus via a sheaf Laplacian, validated through an immunological bastion defense simulation.
Entities (15)
Relation Signals (14)
Strategy Sheaf → correspondsto → Nash Equilibrium
confidence 95% · A global section of Σ is a sheaf-theoretic Nash equilibrium if for every agent v and every alternative τ_v∈Σ(I,v), U_v(τ_v,σ_{-v})≤U_v(σ_v,σ_{-v})
Best-Response Sheaf → models → Nash Equilibrium
confidence 94% · The set of sheaf-theoretic Nash equilibria corresponds to the global sections of the best-response sheaf ℬ
Cohomological Obstruction → classifies → Strategic Consistency Failure
confidence 93% · cohomological obstructions classify failures of strategic consistency.
Game Sheaf → contains → Utility Sheaf
confidence 92% · A game sheaf 𝒢 over 𝒮 consists of: A utility sheaf 𝒰... A strategy sheaf Σ... A reward sheaf ℛ
Game Sheaf → contains → Strategy Sheaf
confidence 92% · A game sheaf 𝒢 over 𝒮 consists of: A utility sheaf 𝒰... A strategy sheaf Σ... A reward sheaf ℛ
Sheaf Laplacian → drives → Geometric Consensus
confidence 91% · The sheaf Laplacian drives consensus: ∂_tϕ=−L_𝒞ϕ
Game Sheaf → contains → Reward Sheaf
confidence 90% · A game sheaf 𝒢 over 𝒮 consists of: A utility sheaf 𝒰... A strategy sheaf Σ... A reward sheaf ℛ
Hybrid Dynamics → combines → Sheaf Laplacian Diffusion
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Abstract
Abstract:The coordination of heterogeneous autonomous agents in dynamic, adversarial environments requires simultaneous satisfaction of geometric constraints, logical consistency, temporal reasoning, and strategic optimization. Existing sheaf- and topos-theoretic frameworks provide powerful tools for geometric consensus, knowledge alignment, and causal planning, but lack explicit models for value, reward, and strategic choice. This report presents a unified categorical framework that integrates event calculus, SCEL-like ensemble formation, and game-theoretic reward structures into a single Grothendieck topos of time-space histories. We introduce the notion of a \emph{game sheaf} whose stalks contain utility functions and policy distributions, and restriction maps encode both parallel transport and best-response dynamics. We prove that Nash equilibria correspond to global sections of a derived best-response correspondence sheaf, while cohomological obstructions classify failures of strategic consistency. A detailed case study of an immunological ``bastion defense'' scenario -- heterogeneous agents forming attack/defense ensembles under resource constraints -- demonstrates the framework's expressiveness. This synthesis provides a rigorous foundation for verifiable, autonomic, and economically rational multi-agent systems.
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- Source: https://arxiv.org/abs/2606.01663v1
- Canonical: https://arxiv.org/abs/2606.01663v1
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A Sheaf Framework for Strategic Multi-Agent Systems: From Consensus to Nash Equilibria Manuel Hernandez Eduardo Sánchez-Soto (May 30, 2026) Abstract The coordination of heterogeneous autonomous agents in dynamic, adversarial environments requires simultaneous satisfaction of geometric constraints, logical consistency, temporal reasoning, and strategic optimization. Existing sheaf- and topos-theoretic frameworks provide powerful tools for geometric consensus, knowledge alignment, and causal planning, but lack explicit models for value, reward, and strategic choice. This report presents a unified categorical framework that integrates event calculus, SCEL-like ensemble formation, and game-theoretic reward structures into a single Grothendieck topos of time-space histories. We introduce the notion of a game sheaf whose stalks contain utility functions and policy distributions, and restriction maps encode both parallel transport and best-response dynamics. We prove that Nash equilibria correspond to global sections of a derived best-response correspondence sheaf, while cohomological obstructions classify failures of strategic consistency. A detailed case study of an immunological “bastion defense” scenario—heterogeneous agents forming attack/defense ensembles under resource constraints—demonstrates the framework’s expressiveness. This synthesis provides a rigorous foundation for verifiable, autonomic, and economically rational multi-agent systems. 1 The Three Headaches of Swarm Coordination Modern multi-agent systems (MAS) operating in open, adversarial environments—such as autonomous defense swarms, robotic search-and-rescue teams, or distributed sensor networks—face three fundamental challenges: a) Geometric coordination: agents must align their physical motions (poses, velocities) under non-holonomic constraints, often without a global coordinate system; b) Logical and temporal consistency: agents must maintain coherent beliefs, plan actions over time, and repair knowledge after unobserved interventions; and, c) Strategic optimization: agents must make decisions that maximize individual or collective utility under scarcity, uncertainty, and potential conflict of interest. Recent advances using sheaf theory, Cartan geometry, and topos semantics have addressed the first two challenges using cellular sheaves [13, 3] and interval temporal planning [8, 9]. However, none of these approaches incorporates value or strategic choice. Agents in real-world scenarios must trade off costs (energy, time, risk) against rewards (territory, resources, mission success). They form ensembles not only based on predicates but also on expected utility. They may cooperate, compete, or defect. This report fills this gap by extending the unified topos with a game-theoretic layer. With this paper, we propose: • A formal integration of event calculus into the temporal site, so that rewards can be updated as actions happen. • A game sheaf whose global sections correspond to Nash equilibria, and where cohomology tells us whether a consistent strategy exists. • A hybrid dynamics that blends sheaf Laplacian diffusion (for consensus) with gradient ascent on expected rewards. • A running example: an immunological bastion defense, where scouts, artillery, and logistics units must cooperate to protect a high-value asset. 2 A Brief Refresher: Sheaves, Topoi, and Why You Should Care We recall the three base frameworks; details can be found in the original references [8, 7, 10]. 2.1 Geometric consensus via cellular sheaves A communication graph G=(V,E)G=(V,E) is equipped with a cellular sheaf C: each vertex v has a stalk (v)=v×ℳvC(v)=K_v×M_v (knowledge and motor), each edge e=u,ve=\u,v\ has a stalk (e)C(e) (interaction space), and restriction maps v⊆eC_v e given by parallel transport (Cartan connection). The sheaf Laplacian drives consensus: ∂tϕ=−Lϕ,(Lϕ)v=∑e∋v⊆e∗(v⊆eϕv−u⊆eϕu). _tφ=-L_Cφ, (L_Cφ)_v= _e vC_v e^*(C_v e _v-C_u e _u). Global sections H0(G;)H^0(G;C) correspond to perfect consensus; obstructions are measured by H1(G;)H^1(G;C). 2.2 Temporal planning and event calculus Time is a poset category T of closed intervals with inclusions. A plan sheaf on T assigns to each interval the set of consistent histories [8]. Actions are natural transformations. Event calculus axioms (inertia, causal consistency) become sheaf conditions [9]. 2.3 The product topos The combined site =×GS=T× G (product category) with product topology yields the topos ()Sh(S). Objects are pairs (I,v)(I,v) with I∈,v∈V(G)I ,v∈ V(G). A sheaf on S assigns stalks containing both temporal histories and geometric/knowledge states. (time intervals)T\;(time intervals)×GT× GG(comm. graph)G\;(comm. graph)Sheaf ℱ on ×GSheaf F on T× Gproductproduct Figure 1: The product site combines temporal intervals and graph vertices. A sheaf on this site encodes both time-dependent and agent-dependent data. 3 Adding Carrots and Sticks: Rewards and Event Calculus 3.1 Reward sheaf A reward is a real-valued fluent. Define the reward sheaf ℛR on S: • For each (I,v)(I,v), ℛ(I,v)=ℝmvR(I,v)=R^m_v (vector of reward components, e.g., health, ammo, score). • Temporal restriction: if no action in I∖I′I I , ρI,I′(r)=r _I,I (r)=r; otherwise add accumulated reward. • Spatial restriction: for edge e=(u,v)e=(u,v), ℛv⊆e:ℛ(v)→ℛ(e)R_v e:R(v) (e) aligns reward estimates (e.g., average or max). The total utility for agent v over interval I is Uv(I)=wv⋅rv(I)U_v(I)=w_v· r_v(I). 3.2 Event calculus as sheaf axioms Following [9], the sheaf conditions encode: 1. Inertia: If no action affecting fluent ϕφ occurs in I, then ϕφ is constant on all subintervals. 2. Causal consistency: If action a initiates ϕφ at time t, then any interval containing t must have ϕφ true after t unless terminated. These are implemented as a sheaf of temporal structures. 4 Game of Sheaves: When Agents Get Strategic 4.1 Definition of a game sheaf A game sheaf G over S consists of: • A utility sheaf U: for each (I,v)(I,v), (I,v)=ℝnvU(I,v)=R^n_v (payoff vector for pure strategy profiles). • A strategy sheaf Σ : for each (I,v)(I,v), Σ(I,v) (I,v) is a compact convex set (mixed strategies). • A reward sheaf ℛR with bilinear pairing ⟨⋅,⋅⟩:Σ(I,v)×(I,v)→ℝ ·,· : (I,v)×U(I,v) . • Restriction maps Σv⊆e _v e that align strategy profiles across edges. A global section of Σ is a family σ=(σI,v)σ=( _I,v) such that for each edge e, Σv⊆e(σI,v)=Σu⊆e(σI,u) _v e( _I,v)= _u e( _I,u) — strategies are consistent across the communication graph. This links our model directly to compositional game theory [6] and compositional modeling of network games [4]. 4.2 Nash equilibrium as a sheaf condition Definition 1. A global section σ of Σ is a sheaf-theoretic Nash equilibrium if for every agent v and every alternative τv∈Σ(I,v) _v∈ (I,v), Uv(τv,σ−v)≤Uv(σv,σ−v),U_v( _v, _-v)≤ U_v( _v, _-v), where utility is computed from (I,v)U(I,v) using restriction maps to align with neighbors. Theorem 1. The set of sheaf-theoretic Nash equilibria corresponds to the global sections of the best-response sheaf ℬB, defined stalkwise by ℬ(I,v)=σv∈Σ(I,v)∣σv∈BRv(σ−v).B(I,v)=\ _v∈ (I,v) _v∈ BR_v( _-v)\. A global section exists if and only if H1(;ℬ)=0H^1(S;B)=0, where ℬB is regarded as a sheaf of sets. This theorem connects strategic equilibrium to sheaf cohomology: non-vanishing H1H^1 indicates that local best-responses cannot be glued into a global equilibrium — a strategic obstruction. ℬ(U)B(U)ℬ(U∩V)B(U∩ V)ℬ(V)B(V)ℬ(U∩V)B(U∩ V)ℬ(U)B(U)ℬ(V)B(V) Figure 2: The gluing condition for best-response sheaf. Local equilibria on overlapping covers must agree; otherwise H1H^1 obstructs a global Nash equilibrium. 4.3 Combined dynamics: diffusion + gradient ascent We propose a hybrid update rule for each agent v at interval I: ϕv(t+1)=ϕv(t)−αLϕv(t)+β∇ϕvUv(ϕ(t)), _v^(t+1)= _v^(t)-α L_C _v^(t)+β _ _vU_v(φ^(t)), where ϕv _v now includes geometric/knowledge components and strategy parameters, L_C is the sheaf Laplacian of the combined sheaf total=geom×know×ΣC_total=C_geom×C_know× , and ∇Uv∇ U_v is the gradient of agent v’s utility with respect to its own strategic variables. Under convexity assumptions (e.g., potential game plus positive semidefinite Laplacian), this dynamics converges to a point that is both a global section (consensus) and a Nash equilibrium. 5 The Bastion Defense: An Immunological Parable Collaborative work in action. We consider a heterogeneous swarm defending a bastion (a high-value asset) against an external attacker. Agents belong to three types: • Scouts: fast, low armor, high sensor range. Role: detect threats, share trajectory predictions. • Artillery: slow, high firepower, limited ammo. Role: engage attackers from distance. • Logistics: medium speed, carries repair/resupply. Role: heal damaged allies, replenish ammo. We will require of the SCEL formalism [12] in the following. Attackers appear at random intervals. Defenders must: (1) maintain geometric formation, (2) share a consistent threat map, (3) dynamically form defense ensembles (SCEL predicates), (4) allocate limited resources to maximize bastion survival probability. 5.1 Agent typologies and immunological metaphor Table 1 maps each agent type to a biological immune counterpart. Table 1: Agent typologies in the immunological bastion defense parable. Agent Type Biological Metaphor Role Armor Speed Sensor Range Scout Dendritic Cell / T-Cell Threat detection, tracking Low High High Artillery B-Cell / Plasma Cell Long-range engagement High Low Low Logistics Helper T-Cell / Macrophage Resupply, repair Medium Medium Medium 5.2 Sheaf construction for the bastion In the bastion defense simulation, bastion HP (health points) represents the remaining structural integrity or survival capacity of the high‑value asset (the bastion) that the swarm of agents must protect. Initial value: 100 HP (for example, as simulated in initState and the Python/Elixir versions). Decrease: When an enemy reaches the bastion (enemyHitsBastion), the bastion loses HP equal to the enemy’s damage (typically 8 points per hit). Increase: Logistics agents can repair the bastion when they are close (dist << 60) and the bastion HP is below 100. Each repair action restores up to 10 HP (but never above 100). Game over: When bastion HP falls to 0 or below, the simulation displays “BASTION LOST” and waits for a reset (R key). Thus, bastion HP is the core success metric: keeping it above zero while enemies attack is the primary objective of the defending agents. For each agent v and time interval I, the stalk total(I,v)C_total(I,v) is a product: total(I,v)=geom(I,v)×know(I,v)×Σ(I,v)×ℛ(I,v),C_total(I,v)=C_geom(I,v)×C_know(I,v)× (I,v)×R(I,v), where: • geomC_geom: motor MvM_v (pose in SE(2)SE(2)). • knowC_know: threat positions, bastion HP, ammo count. • Σ(I,v) (I,v): distribution over actions attack, defend, resupply, repair. • ℛ(I,v)R(I,v): reward components (damage dealt, damage taken, ammo used, repair given). Restriction maps: • Temporal: propagate knowledge forward (inertia), accumulate rewards. • Spatial (edge e=(u,v)e=(u,v)): Cartan parallel transport aligns poses; threat positions transformed to common frame; strategies averaged; reward estimates aligned. 5.3 Event calculus rules for the scenario The following components allow us to formulate an instance of the event calculus: Fluents: underAttack(bastion), hasAmmo(v), inRange(v, threat), alive(v). Actions: fire(v,t), move(v,waypoint), repair(v,w), resupply(v,w). Sample axioms: initiates(fire(v,t), damage(t), T)terminates(fire(v,t), hasAmmo(v), T) initiates(resupply(v,w), hasAmmo(v), T)holdsAt(bastionHP = 0, T) → terminal(T) Reward at interval I for agent v: Rv(I)=∑t∈Iγt(damageDealt(t)−c1⋅damageTaken(t)−c2⋅ammoCost(t)+c3⋅repairGiven(t)),R_v(I)= _t∈ Iγ^t (damageDealt(t)-c_1·damageTaken(t)-c_2·ammoCost(t)+c_3·repairGiven(t) ), with discount factor γ∈(0,1)γ∈(0,1). 5.4 Cognitive immune pathways for threat integration Scout agents implement a decision pipeline (Table 2) to filter sensor data. Table 2: Cognitive Immune System (CIS) decision pathways. Pathway Condition Action Accept m>0m>0, λ>1λ>1, Hknow1=0H^1_know=0 Merge threat, broadcast coordinates Reject m<0m<0 or violates kinematics Drop data, reduce source trust Quarantine m≈0m≈ 0 (critical state) Isolate, send scouts for verification Reframe Hknow1≠0H^1_know≠ 0 but neighbors confirm Escalate to human-in-the-loop 5.5 Distributed asynchronous algorithm Each agent runs an asynchronous loop: 1. Sense local environment: update knowledge stalk. 2. Communicate with neighbors: exchange state (ϕvgeom,ϕvknow,σv,rv)(φ^geom_v,φ^know_v, _v,r_v). 3. Compute inconsistency: δuv=v⊆e(ϕv)−u⊆e(ϕu) _uv=C_v e( _v)-C_u e( _u). 4. Update using hybrid dynamics: ϕv←ϕv−α∑uv⊆e∗δuv+β∇σvUv _v← _v-α _uC_v e^* _uv+β _ _vU_v. 5. Sample action from updated strategy σv _v and execute. 5.6 Obstruction analysis Three types of failures detected cohomologically: 1. Geometric obstruction: non-zero holonomy around a cycle (Hgeom1≠0H^1_geom≠ 0). 2. Logical obstruction: incompatible threat maps (Hknow1≠0H^1_know≠ 0). 3. Strategic obstruction: no global section of the best-response sheaf (Hgame1≠0H^1_game≠ 0). A Künneth-type formula decomposes the total obstruction: Htotal1≅Htime1⊗Hgraph0⊕Htime0⊗Hgraph1⊕torsion.H^1_total H^1_time H^0_graph\; \;H^0_time H^1_graph\; \;torsion. Thus designers can diagnose whether failure is due to timing, graph connectivity, or logical/strategic inconsistency. 6 Algorithmic Realization of Strategic Consensus This section details the explicit, step-by-step algorithms governing the multi-agent swarm’s spatial alignment, strategic learning, cohomological diagnostics, and logical threat evaluation. 6.1 Algorithm 1: Partially Asynchronous Nonlinear Sheaf Diffusion This algorithm drives spatial and belief consensus across heterogeneous agent stalks in environments plagued by computation and communication delays [2]. 1. Initialization: • Let the communication network be modeled as an undirected graph G=(V,E)G=(V,E). • For each agent i∈Vi∈ V, initialize its state vector i(0)∈ℱ(i)=ℝdix_i(0) (i)=R^d_i on the vertex stalk. • For each incident vertex-edge pair i⊴iji ij, define the linear restriction map ℱi⊴ij:ℱ(i)→ℱ(ij)F_i ij:F(i) (ij) projecting local states into the shared interaction space ℱ(ij)=ℝdijF(ij)=R^d_ij. • Assign a strongly convex edge potential function Uij:ℱ(ij)→ℝU_ij:F(ij) representing local agreement criteria [1]. 2. Asynchronous Execution Loop: • For each agent i∈Vi∈ V, let Ti⊆0,1,2,…T_i \0,1,2,…\ be the set of discrete times at which agent i updates its state. • Let B≥0B≥ 0 represent the maximum delay bound. At any time t∈Tit∈ T_i, agent i retrieves the delayed state of neighbor j∈Nij∈ N_i, denoted as j(τji(t))x_j( _j^i(t)) where t−B≤τji(t)≤t-B≤ _j^i(t)≤ t. • At time step t∈Tit∈ T_i, agent i executes: (a) Compute Mismatch: Calculate the local projection difference ij(t)∈ℱ(ij)y_ij(t) (ij) across each incident edge e=ije=ij: ij(t)=ℱj⊴ij(j(τji(t)))−ℱi⊴ij(i(t))y_ij(t)=F_j ij(x_j( _j^i(t)))-F_i ij(x_i(t)) (b) Evaluate Edge Gradients: Compute the force interaction vector Φij(t)=∇Uij(ij(t)) _ij(t)=∇ U_ij(y_ij(t)). (c) Local State Update: Apply the adjoint restriction maps ℱi⊴ij∗F_i ij^* and step size αi>0 _i>0 to advance its state: i(t+1)=i(t)+αi∑j∈Niℱi⊴ij∗(Φij(t))x_i(t+1)=x_i(t)+ _i _j∈ N_iF_i ij^*( _ij(t)) • For all t∉Tit∉ T_i, the state is held constant: i(t+1)=i(t)x_i(t+1)=x_i(t). 6.2 Algorithm 2: Hybrid Strategic Consensus and Expected-Utility Update This algorithm executes the combined update rule, blending physical coordinate/belief consensus with gradient-based policy updates to guide agents toward global Nash equilibria [8, 9]. 1. Augmented State Initialization: • For each agent i∈Vi∈ V, construct the augmented local state ϕi(t)=(ϕigeom(t),ϕiknow(t),σi(t)) _i(t)=( _i^geom(t), _i^know(t), _i(t)) lying in the total stalk space total(I,i)C_total(I,i). • Let σi(t)∈Σ(I,i) _i(t)∈ (I,i) be a probability distribution over the discrete action space A=attack,defend,resupply,repair.A=\attack,defend,resupply,repair\\ . 2. Distributed Execution: At each local iteration time t: (a) Query Delayed Neighbors: Read neighboring state vectors ϕj(tdelayedi)=(ϕjgeom(tdelayedi),ϕjknow(tdelayedi),σj(tdelayedi)) for j∈Ni. _j(t^i_delayed)=( _j^geom(t^i_delayed), _j^know(t^i_delayed), _j(t^i_delayed)) for j∈ N_i\ . (b) Evaluate Sheaf Mismatch: Compute the total local sheaf Laplacian vector: (Lϕ(tdelayedi))i=∑j∈Nii⊴ij∗(i⊴ij(ϕi(t))−j⊴ij(ϕj(tdelayedi)))(L_Cφ(t^i_delayed))_i= _j∈ N_iC_i ij^* (C_i ij( _i(t))-C_j ij( _j(t^i_delayed)) ) (c) Compute Utility Gradient: Calculate the local expected-payoff gradient i(t)∈ℝ|A|g_i(t) ^|A| with respect to the agent’s strategy parameters: i(t)=∇σiUi(σi(t),σ−i(tdelayedi))g_i(t)= _ _iU_i( _i(t), _-i(t^i_delayed)) (d) State Transition: Apply the joint update step with consensus rate αi>0 _i>0 and strategic rate βi>0 _i>0: ϕi(t+1)=ϕi(t)−αi(Lϕ(tdelayedi))i+βi[i(t)] _i(t+1)= _i(t)- _i(L_Cφ(t^i_delayed))_i+ _i bmatrix0\\ 0\\ g_i(t) bmatrix (e) Action Sampling: Sample a concrete action ai(t)∼σi(t+1)a_i(t) _i(t+1) for immediate physical execution. 6.3 Algorithm 3: Topological Cohomology and Obstruction Diagnostics This diagnostic algorithm is executed globally or over localized subgraphs to detect, classify, and isolate geometric or strategic coordination failures [8, 9]. To this end, we have a remark on the Moore–Penrose pseudoinverse to be used in Algorithm 3, to be shown below. The global coboundary matrix D is constructed from the sheaf’s restriction maps. Concretely, let the vertex stalks have total dimension d0=∑v∈Vdimℱ(v)d_0= _v∈ V (v) and the edge stalks total dimension d1=∑e∈Edimℱ(e)d_1= _e∈ E (e). Then D∈ℝd1×d0D ^d_1× d_0 acts on a tuple of vertex assignments ∈ℝd0x ^d_0 by sending it to the tuple of edge-wise differences (ℱv⊴e(xv)−ℱu⊴e(xu))e=(u,v) (F_v e(x_v)-F_u e(x_u) )_e=(u,v). Its Moore–Penrose pseudoinverse D†D is the unique matrix satisfying DD†D=D,D†DD†=D†,(DD†)T=DD†,(D†D)T=D†D.D D=D, D D =D , (D )^T=D , (D D)^T=D D. In the context of the diagnostic algorithm, D†D serves two purposes: 1. Minimum‑norm solution of inconsistent systems: Given a boundary mismatch vector ∈ℝd1b ^d_1 (e.g., discrepancies measured on edges), the vector ∗=−D†z^*=-D b is the unique minimiser of ‖D+‖2\|Dz+b\|_2 with smallest Euclidean norm. This yields the “diffused” vertex states that best fit the observed edge mismatches. 2. Harmonic projection: The matrix H=I−DD†H=I-D projects any edge vector onto kerDT D^T, which is exactly the space of sheaf cocycles that are not coboundaries – i.e., the representatives of the first sheaf cohomology group H1(;ℬ)H^1(S;B). Consequently, ∗=Hr^*=Hb is non‑zero precisely when the inconsistencies b contain a cohomological obstruction that prevents gluing local data into a global section (a global Nash equilibrium or consensus state). The support of ∗r^* identifies the offending edges or cycles. Thus computing D†D turns a purely topological obstruction (cohomology) into a numerically computable residual, enabling real‑time diagnosis of geometric, logical, or strategic failures in the multi‑agent system. This is the Algorithm 3: 1. Cochain Complex Construction: • Compile the global coboundary matrix D∈ℝd1×d0D ^d_1× d_0 mapping the disjoint union of vertex stalks to the disjoint union of edge stalks, where d0=∑v∈Vdimℱ(v)d_0= _v∈ V (v) and d1=∑e∈Edimℱ(e)d_1= _e∈ E (e). • Define the target boundary mismatch or excitation vector ∈C1(G;ℱ)b∈ C^1(G;F). 2. Projector Computation: • Compute the Moore-Penrose pseudoinverse D†D of the global coboundary matrix. • Construct the harmonic projection matrix H∈ℝd1×d1H ^d_1× d_1 onto kerDT D^T: H=I−DD†H=I-D • Construct the diffusive operator G=D†G=D . 3. Obstruction Locus Isolation: • Compute the optimal minimum-norm residual edge force vector: ∗=Hr^*=Hb • Compute the converged internal vertex activations (diffused states): ∗=−Gz^*=-Gb • Evaluate Cohomology Class: – If ∗=r^*=0, then H1(;ℬ)=0H^1(S;B)=0. Local states can be seamlessly glued into a globally stable Nash equilibrium or geometric section. – If ∗≠r^* 0, then H1(;ℬ)≠0H^1(S;B)≠ 0. Isolate the coordinates e∈Ee∈ E where e∗≠0r^*_e≠ 0. This identifies the exact network loops or conflicting agent boundaries causing the strategic or geometric obstruction. 6.4 Algorithm 4: Cognitive Immune System (CIS) Threat Integration This logic-based decision pipeline runs on Scout agents to filter, classify, and securely merge newly observed targets into the global threat map [5]. 1. Threat Perception: Capture a candidate observation trajectory obsx_obs over a temporal interval I∈I . 2. Coherence and Stability Evaluation: • Temporarily inject obsx_obs into the local belief sheaf know(I,v)C_know(I,v). • Calculate the local first knowledge-cohomology group Hknow1H^1_know on the overlapping neighborhood coverage. • Estimate the resulting coherence margin m∈ℝm and stability parameter λ∈ℝλ of the belief state. 3. Pathway Routing Rule: • Accept: if m>0m>0, λ>1λ>1, and Hknow1=0H^1_know=0 → merge threat, broadcast coordinates. • Reject: if m<0m<0 or violates kinematics → drop data, reduce source trust. • Quarantine: if m≈0m≈ 0 (critical state) → isolate, send scouts for verification. • Reframe: if Hknow1≠0H^1_know≠ 0 but neighbors confirm → escalate to human-in-the-loop. 7 Computational Realization, Complexity, and Formal Verification The framework is computationally realizable using existing libraries for cellular sheaves (e.g., SheafPy) extended with game-theoretic solvers (fictitious play, replicator dynamics). We have carried out some graphical simulations in Pygame/Python. Realistic simulations have also programmed in Erlang and Elixir. Table 3 summarizes complexity. Table 3: Algorithmic complexity of the hybrid strategic swarm architecture. Component Complexity Critical Parameters Scalability Sheaf Laplacian update O(|E|⋅d2)O(|E|· d^2) d: stalk dimension (<100<100) Linear in local degree Gradient ascent O(|V|⋅|A|)O(|V|·|A|) |A||A|: actions (4–10) Highly parallel Cohomology diagnostic O((|V|+|E|)3)O((|V|+|E|)^3) Global (run occasionally) Acceptable for diagnosis Asynchronous convergence results apply as long as the reward gradient is Lipschitz [2]. Verification uses the internal logic of the topos; properties like “bastion survives until time T” correspond to global sections of a subobject. 8 Related Work Our synthesis builds directly on the three pillars: sheaf-theoretic planning [8], SCEL-like coordination [10], and event calculus as a topos [9]. The addition of game theory connects to distributed potential games [11] and sheaf learning [1]. The immunological bastion metaphor is reminiscent of artificial immune systems [5], but here formalized categorically. 9 Conclusion and Future Work We have presented a unified categorical framework that integrates geometric consensus, logical planning, temporal reasoning, and strategic optimization in a single Grothendieck topos. Key innovations: • A reward sheaf and event-calculus semantics layered on the temporal site. • Game sheaves whose global sections correspond to Nash equilibria, with cohomological obstructions classifying strategic failures. • A hybrid dynamics combining sheaf Laplacian diffusion and utility gradient ascent. • A detailed case study showing how heterogeneous agents self-organize attack/defense ensembles under resource constraints. Table 4 summarizes the core technical advancements. Table 4: Explicit contributions of the unified sheaf-theoretic game topos framework. Contribution Category Technical Mechanism Formal Foundation Benefit Geometric-Temporal Integration Product site =×GS=T× G Grothendieck Topos Theory Unifies space and time Strategic Localization Game sheaf G, strategy sheaf Σ Compositional Game Theory Models utility over networks Obstruction Diagnostics Best-response sheaf ℬB, cohomology H1H^1 Sheaf Cohomology Detects strategic failures Decentralized Convergence Hybrid Laplacian + gradient dynamics Nonlinear Sheaf Diffusion Converges to consensus + equilibrium Future work includes: (i) implementing several prototypes and variants in Elixir; (i) extending the game sheaf to Bayesian games using hypergraph sheaves; (i) applying the framework to real-world swarm robotics. We can assert that the theoretical foundation is now laid for a truly autonomic, economically rational, and formally verifiable multi-agent intelligence. References [1] N. Anwer, H. Riess, and M. Hale (2026) Multi-agent system identification with nonlinear sheaf diffusion. arXiv arXiv:2605.11204. Cited by: 4th item, §8. [2] Asynchronous Nonlinear Sheaf Diffusion for Multi-Agent Coordination (2026) Asynchronous nonlinear sheaf diffusion for multi-agent coordination. arXiv arXiv:2510.00270. Cited by: §6.1, §7. [3] J. Curry (2014) Sheaves, cosheaves and applications. Ph.D. Thesis, University of Pennsylvania. Cited by: §1. [4] J. di Lavore, M. Román, and J. Hedges (2021) Compositional modelling of network games. In CSL 2021, LIPIcs. Cited by: §4.1. [5] S. Forrest, A. S. Perelson, L. Allen, and R. Cherukuri (1994) Self-nonself discrimination in a computer. In Proceedings of IEEE Symposium on Research in Security and Privacy, Cited by: §6.4, §8. [6] J. Hedges (2016) Compositional game theory. arXiv arXiv:1603.04641. Cited by: §4.1. [7] M. Hernandez and E. Sanchez-Soto (2026) An explanation topos: requesting knowledge provenance and consistency in multi-agent systems. In 2026 International Conference on Artificial Intelligence, Computer, Data Sciences and Applications (ACDSA), p. 1–6. Cited by: §2. [8] M. Hernandez and E. Sanchez-Soto (2026) Sheaf-theoretic planning: a categorical foundation for resilient multi-agent autonomous systems. arXiv arXiv:2605.01879. Cited by: §1, §2.2, §2, §6.2, §6.3, §8. [9] M. Hernandez and E. Sanchez-Soto (2026) Topos-theoretic formalism for causal event calculus. In Congreso Mexicano de Inteligencia Artificial, Communications in Computer and Information Science. Note: to appear Cited by: §1, §2.2, §3.2, §6.2, §6.3, §8. [10] M. Hernandez and E. Sanchez-Soto (2026) Topos-theoretic framework for verifiable multi-agent coordination and team formation. In 2026 ACDSA, p. 1–6. Cited by: §2, §8. [11] J. R. Marden and J. S. Shamma (2014) Game theory and distributed control. In Handbook of Game Theory, Vol. 4. Cited by: §8. [12] R. D. Nicola, M. Loreti, R. Pugliese, and F. Tiezzi (2014) A formal approach to autonomic systems programming: the scel language. ACM TAAS 9 (2). Cited by: §5. [13] M. Robinson (2014) Sheaves, cosheaves and applications. arXiv arXiv:1303.3255. Cited by: §1.