6G Reasoning Agent Societies: Composition, Scaling, and Consistency-Time Laws

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6G Reasoning Agent Societies: Composition, Scaling, and Consistency-Time Laws

arXiv:2610.34357v1 [cs.NI] 25 Oct 2026

Hatim Chergui, Senior Member, IEEE, Carolina Fernández-Martı́nez, Mustapha Benjillali, Senior Member, IEEE, Mehdi Bennis, Fellow, IEEE, and Merouane Debbah, Fellow, IEEE

Abstract—Emerging 6G control planes would compose Large Language Model (LLM) agents that exchange interpreted conclusions, so hallucinations can propagate even over error-free transport. Recent architectures audit each message locally against the sender’s observation mask, but isolated audits miss collective behavior. Modeling the agent society as a cellular sheaf, we establish three laws valid on arbitrary topologies. First, the composition law: network-wide divergence exceeds the sum of per-agent audit scores by exactly their hidden total correlation. This blind spot masks up to 23% of joint divergence and, because the declared signal cancels in the cognitive signal-to-noise ratio (SNR), overstates it by 1.14 dB purely through topology; the resulting trust inflation preserves valid consensus states but slows convergence. Second, the scaling law: reconciliation decouples across metric counters, so the worst-connected dimension sets the pace. Consistency time grows for a society size n as Ω(n2/d ) on polynomial-growth topologies of degree d, and paths breach a 1 s near-real-time budget at n∗ = 13 agents, whereas expanders (Θ(1)) and radius-3 local audits (O(1), ≈ 300 ms) stay bounded. Third, the consistency-time law: the timing error from omitted correlations is one-sided and capped by the worst-measured link. These laws, validated on simulated topologies and deployed LLM societies, give provable worst-case latency ceilings for near-realtime operation. Index Terms—6G, agentic systems, cellular sheaves, cognitive SNR, large language models, O-RAN, theory of mind, topology, trust.

I. I NTRODUCTION

P

ROPOSALS for next-generation O-RAN architectures are rethinking the near-real-time RAN Intelligent Controller (RIC). Instead of rigid, rule-based xApps, these networks employ LLM agents that do more than execute code: they interpret E2 measurement reports, form internal beliefs about the state of the network’s cells, and negotiate resource allocation in natural language before acting [1], [2]. This leap in capability introduces a blind spot. Because the agents are proprietary black boxes, an operator cannot audit their internal logic to verify how a conclusion was reached, and the control plane is no longer transmitting bits reliably but conveying interpreted meaning [3], [4]. Protection has to be redefined

H. Chergui and C. Fernández-Martı́nez are with the i2CAT Foundation, Spain. (e-mails: chergui@ieee.org, name.surname@i2cat.net) M. Benjillali is with INPT, Morocco. (e-mail: benjillali@ieee.org). M. Bennis is with the University of Oulu, Finland (e-mail: mehdi.bennis@oulu.fi). M. Debbah is with the Research Institute for Digital Future, Khalifa University, 127788 Abu Dhabi, UAE (e-mail: merouane.debbah@ku.ac.ae). This work was supported by grants DGR-2026-2-DIGITALSHIELD, funded by the Research and Universities Department of the Catalonian Government as well as by COALESCE-6G PID2024-163028OB-I00, funded by MICIU/AEI/10.13039/501100011033/FEDER, EU.

accordingly. In a classical control plane, resilience is transport integrity: a packet delivered intact is correct. In an agentic one, a perfectly delivered conclusion can still be a hallucination inferred from misunderstood or unmonitored metrics, so the network must secure trust rather than integrity [5], [6], which means endowing it with a machine Theory of Mind—the ability to judge an agent’s credibility from its behavior and interactions [7], [8]. A. Literature Review Four lines of work bear on that problem and none of them closes it. The first is the agentic control plane itself. The near-realtime RIC, its E2 interface and the xApp model that LLM agents are proposed to inhabit are well characterized [9], and multi-agent frameworks show that agents given roles, memories and a protocol do coordinate on non-trivial tasks [1], [2], [4]. The same literature establishes the failure mode that motivates auditing: a language model produces fluent, confident statements unsupported by its inputs [10], and at the receiver those are indistinguishable from well-founded ones. Outputs are nevertheless judged one at a time, which is the assumption this paper prices. The second is semantic and goal-oriented communication, which moves the design objective from bit fidelity to the fidelity of conveyed meaning and gives an information-theoretic account of what a receiver needs for its task [3], [6], [11]. It says what should be preserved end to end, not what an evaluator of meaning can and cannot see—a different question, and the one a trust layer faces. The third is trust and reputation. Scores are accumulated per interaction and aggregated by sums, averages or Bayesian updates [12], and in agentic control planes each message is scored against an expectation reconstructed from the sender’s public observation mask [5]. The locality is not an oversight but a necessity, since a near-real-time loop cannot run joint inference over message histories. Every one of these aggregation rules, however, is exact only when the evidence being combined is independent—and circulating conclusions introduce dependence. Fourth, the spectral theory of agreement models network consensus as a linear diffusion process, ẋ = −Lx, where nodes iteratively average their states with neighbors [13], [14], [15]. The rate of this diffusion follows a 1/λ2 relaxation law, meaning the graph Laplacian’s spectral gap λ2 controls how fast differences dissipate. Consequently, the edge weights of

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L can be treated as tunable design parameters and optimized directly to maximize convergence speed [16]. Cellular sheaves generalize this to agents holding different quantities that agree only where their vocabularies overlap [17], and sheaf diffusion has modeled disagreement among agents whose private frames differ [18]. Classical information theory supplies the measures: Kullback–Leibler geometry [19], total correlation as the multivariate extension of mutual information [20], and the composition results separating single-link from cascaded capacity [21], [22].

s UP·MOB

u CP·MOB link e

w UP·CP·MOB

v UP·MOB

Three gaps remain. First, the composition results of network information theory govern transport: what a cascade of channels can carry, not what a cascade of evaluations can detect. With no counterpart for credibility, a trust layer that sums perlink scores inherits the sum’s independence assumption, and with it a cost no better expectation can remove. Second, consensus theory prices agreement for agents holding the same state but disagreeing about its value. Sheaf Laplacians [17], [18] handle different vocabularies, but ask whether opinions converge, and to what—not how long that takes under the topology the subscriptions induce, which is what a near-real-time budget is spent on. Third, the weights here are measurements, not design variables. Where [16] chooses weights to make a network fast, a trust layer reads them from scores that Theorem 1 in this paper shows to be optimistic, then derives its convergence time from them. A controller sizing a budget on that number trusts a self-report whose error the literature does not bound. We close these gaps with three laws, each stated for an arbitrary topology, proved from the architecture and validated without fitted parameters: • The composition of trust (Section III). Link-by-link evaluation underestimates the society’s divergence by exactly the total correlation of its messages, for any topology and per-agent expectations. • Scaling and consistency time (Section IV). A society reconciles as slowly as its worst-connected counter, so topology alone fixes consistency time: Ω(n2/d ) under polynomial growth, Θ(1) on expanders, and O(1) within a fixed neighborhood. • Bounding the reported consistency time (Section IV). The reported consistency time never exceeds the true one and falls short of it by at most the worst-audited link’s inflation, so near-real-time budgets can be sized on it safely. • Validation (Section V). The laws are tested on simulated sheaves swept over topology and size, and on LLM agent societies prompted with O-RAN measurement reports. II. S YSTEM M ODEL A. Agents, Masks, Messages, and Beliefs The control plane operates as a society of agents connected over a graph G = (V, E), shown in Fig. 1; N (v) denotes the neighbors of agent v. Each agent subscribes to a specific

one message per agent mv is carried by every link at v one expectation per sender qv is built from Av alone, whoever receives

t CP·MOB

trust layer: one weight per link

we

1

we

2

$$N_{\text{local}}=\sum_v D_{\text{KL}}\left(p_v \|\ q_v\right)$$

we

3

$$\Phi \geq 0$$

we

we

4

5

$$N_{\mathrm{joint}}=D_{\text{KL}}(p_V\|\ \prod_{v}q_v)$$

the gap is the total correlation of the messages — Theorem 1, on any topology; the layer never scores it E2SM-KPM reports

B. Gaps and Contributions

masks Av: public O1 metadata

near-RT RIC: LLM xApps on an arbitrary graph G = (V, E)

E2 control / policy

6G RAN — E2 nodes (gNB-CU / gNB-DU), cells, slices

Fig. 1. A society of LLM xApps on a near-real-time RIC. The agents interact over graph G = (V, E), each monitoring the RAN through a public mask Av . The trust layer scores each link against an expectation built from the sender’s mask alone.

set of E2 counters, forming its public observation mask Av , and observes the network metrics ov = θ|Av for a given cell state θ. Agents score the cell over a shared vocabulary C of scoring dimensions c—in the running example, userplane load (UP), control-plane signaling (CP), and mobility stress (MOB)—each fed by one or more E2 counters; Cv ⊆ C denotes the dimensions covered by Av . Each round, agent v emits a message over the whole vocabulary, produced by its proprietary policy πv from its own observation and the messages its neighbors sent in the previous round. Its belief is the part of that message its mask backs:
mv = πv ov , {mu }u∈N (v) ∈ [0, 100]C , xv = mv Cv . (1) Each entry of xv is a score on the common 0–100 scale, the 2 on dimenmean of a belief distribution with variance σv,c −2 )c∈Cv . sion c; the belief’s precision matrix is Λv = diag(σv,c The remainder of the message, mv |C\Cv , is asserted without evidence. The message is indexed by vertex because the same vector traverses every incident link. Because masks are public O1 metadata, an operator can invert a generic emission policy to calculate an expected belief x̂v on the same dimensions, based strictly on what v is allowed to see—its observation ov , with no neighbor input. Crucially, asking agents with asymmetric views to agree on everything is ill-posed. If a signaling-focused agent lacks user-plane counters and a user-plane agent lacks controlplane counters, their beliefs do not even range over the same dimensions, and forcing xu = xv would require both to guess blindly on unmonitored metrics. Consensus is meaningful only on the projection onto mutually observable metrics. The mathematical framework that formalizes this kind of localized, overlapping agreement is a cellular sheaf [17]. B. A Running Example To illustrate why partial agreement is necessary and how the trust layer evaluates it, consider the following scenario. Example 1 (Two agents and a signaling storm). Two agents monitor a single cell: Signaling Watch (u, for the control

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plane) and Slice Guardian (v, for the user plane); both also monitor mobility. Their masks Au and Av capture data during a signaling storm occurring over a lightly loaded user plane. Mask

E2SM-KPM counter

Dimension

Au

RACH.PreambleDedCell MM.HoExeFailCell RRU.PrbTotDl DRB.UEThpDl MM.HoExeFailCell

CP MOB UP UP MOB

Av

δx = (5, −2 | 8, 1).

Reading 2740 46 32 51 46

Agent u emits the message mu = (70, 85, 55) over (UP, CP, MOB). Agent v, after reading it, emits mv = (52, 60, 58), even though its own counters would have put the user plane near 35. By (1), mu restricts to the belief xu = (−, 85, 55), and mv to xv = (52, −, 58). The message mu is a single point; the law pu is the statistical distribution this message follows across operating conditions—it should be sharp on CP but diffuse on UP—and the expectation law qu is what the trust layer builds from the mask alone. Agent u’s UP score of 70 is a confident assertion its mask cannot support, and the trust layer rightfully penalizes it. Agent v’s UP score of 52, by contrast, passes the audit because its mask covers UP—yet that score was heavily influenced by reading u’s hallucinated message. No local inspection against qv can detect this flaw: the provenance of the error is hidden entirely in the statistical dependence between the agents. C. The Trust Sheaf To manage these overlapping views mathematically, we model the network as a trust sheaf. Extending the example, u still tracks CP and MOB, v tracks UP and MOB, and a third agent z tracks all three dimensions; in this configuration u and v are each linked to z but not to each other. Stalks and shared vocabularies. A sheaf F assigns each agent a vertex stalk F (v), the space of its beliefs over Cv , and each link an edge stalk F (e), the dimensions its two endpoints share. Before communicating across a link, an agent applies a restriction map Fv→e : F(v) −→ F(e),

v → e,

(2)

which acts as a filter, dropping every dimension its neighbor does not track. Defining consensus. Consensus on a link e = {u, z} means the agents’ filtered beliefs match: Fu→e xu = Fz→e xz . If u’s belief is (80, 57) and z’s is (44, 80, 57), they agree on their link: u’s (80, 57) equals z’s restriction to the shared CP and MOB, and u’s blindness to z’s UP score of 44 does not break the agreement. Measuring disagreement. When agents disagree, their current beliefs are stacked into a global state vector, a 0-cochain L x ∈ F(v). For the three agents, with u and v holding v their beliefs from Example 1:
(3) x = −, 85, 55 52, −, 58 44, 80, 57 . | {z } | {z } | {z } u

v

disagreement is (85 − 80, 55 − 57) = (5, −2); between v and z (sharing UP, MOB) it is (52 − 44, 58 − 57) = (8, 1). The resulting 1-cochain collects these residuals:

z

The coboundary operator δ subtracts the restricted beliefs across each link. Between u and z (sharing CP, MOB) the

(4)

The Laplacian and relaxation. The network penalizes these residuals in proportion to the trust weight we of each link, defined in Section II-D. With W the diagonal matrix of link weights, the sheaf Laplacian LF ≜ δ ⊤ Wδ yields their weighted sum: X x⊤ LF x = we ∥(δx)e ∥2 . (5) e

To resolve conflicts, the network undergoes relaxation, ẋ = −LF x: a local gradient descent that moves each agent’s beliefs toward its trusted neighbors, and stops only when every shared dimension agrees and δx = 0. Gluing. Gluing is the property that lets local agreements cascade into network-wide consensus. An assignment of beliefs is consistent when all differences across links vanish, δx = 0; such a state is a global section, an element of H 0 (G; F) = ker δ. Anchoring z’s beliefs, an audit of the two links to z alone drives xu = (80, 57) and xv = (44, 57), and so brings u and v, which share no link, to a common MOB = 57. On a connected graph, pairwise agreement stitches itself into global agreement, which is what licenses a purely local audit. Because W is positive diagonal, ker LF = ker δ = H 0 whatever the weights: trust sets how fast the society reaches a global section, never which global sections exist. If calibrations conflict around loops, consensus contracts to the consistent subspace H 0 , and relaxation projects the initial beliefs onto it with disagreement decaying as e−λ2 t [13], where λ2 is the smallest nonzero eigenvalue of LF . The time for disagreement to fall to a fraction ε of its initial value is the consistency time τ (F ) ≜ λ2 (LF )−1 log(1/ε).

(6)

D. The Cognitive SNR Receivers evaluate trust via a machine Theory of Mind, auditing whether a sender’s evidence supports its claims. Comparing a transmitted message against an expectation derived from the sender’s public mask yields a cognitive signal-tonoise ratio (SNR) [5]: Nv→e ≜ DKL (Fv→e xv ∥ Fv→e x̂v ) ,

γv→e ≜

Sv , Nv→e (7)

w(γ) ≜ 1 − exp(−ηγ) ∈ [0, 1), where, as throughout, a divergence is taken between the belief distributions whose means these vectors hold. The noise Nv→e penalizes hallucinations by measuring how far the sender’s restricted message departs from the expectation its mask supports. The signal Sv is the confidence the sender asserts, taken as in the companion formulation [5] to be the trace of its belief’s precision matrix:
X 1 Sv ≜ Tr Λv = . (8) 2 σv,c c∈Cv

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Since scores are pure numbers on the 0–100 scale, Sv is a dimensionless precision, whereas Nv→e is a divergence in bits; the ratio γv→e therefore carries units of inverse bits, reading as asserted precision per bit of unexplained divergence. It is reported against a declared reference γ0 = 1 bit−1 , exactly as power is reported in dBm against 1 mW, and written in dBcog : a value of x dBcog means 10 log10 (γv→e /γ0 ) = x. Nothing in the three laws below turns on that choice: each is stated as a ratio in which Sv cancels, so the reference cancels with it. The ratio γv→e acts as a native hallucination detector: asserting high confidence without observational support spikes Nv→e and collapses the SNR. Finally, η scales γ into a normalized weight, and reconciliation across a link is bottlenecked by its least credible participant: we ≜ w(γe ) with γe ≜ minv→e γv→e .

Q P where Φ(mV ) ≜ DKL pV = v∈V pv v∈V h(mv ) − h(mV ) and h(·) is the differential entropy. Q Proof: Assume pV ≪ v qv and all terms finite. This holds in practice because each qv , being the law of a generic policy’s output, assigns positive density to every score on every dimension—on dimensions outside the mask it reduces to an uninformative prior—so Q no feasible message is declared impossible. Inserting log v pv ,   pV Njoint ≜ Ep log Q v qv  Q    pv pV v = Ep log Q +Ep log Q . (11) pv qv | {z v } | {z v } (a)

(b)

Q

E. Message Laws and the Two Divergences Across operating conditions θ, (1) makes each mv a random vector. The ensemble mV ≜ (mv )v∈V follows a joint law pV with marginals pv , and its dependence has two sources: shared counters feed a common θ into several observations, and (1) feeds each agent its neighbors’ messages. The trust layer’s expectation qv is the law of the message a generic policy would emit from ov alone, with no neighbor input. Agent v contributes Nv ≜ DKL (pv ∥ qv ), taken over the whole vocabulary so that unbacked claims are charged, and the reported and joint divergences are X
Q Nlocal ≜ Nv , Njoint ≜ DKL pV v∈V qv . (9) v∈V

Both measure cognitive noise in bits. In Example 1, Nlocal charges u for its unbacked UP claim and clears v, whose UP score is backed by its own counters; Njoint also charges for v’s UP score moving with u’s, a dependence no marginal can represent. Assumption 1 (Per-agent expectation). The reference law against which Q the trust layer scores a relayed conclusion is the product v∈V qv , each factor built from its agent’s own observation alone. III. T HE C OMPOSITION L AW Near-real-time control loops must evaluate messages link by link, because joint audits over full network histories are computationally infeasible. Yet as agents read peer messages, their conclusions become statistically intertwined. Local audits measure individual deviations from expectation but cannot detect collective reinforcement. The composition law formalizes this blind spot: localized auditing underestimates the joint divergence by exactly the hidden dependence among agents, and so inflates reported trust. Theorem 1 (Society divergence). For any finite society and any interaction topology, the locally visible and joint divergences of (9) differ by exactly the total correlation Φ of the messages mV : Njoint = Φ(mV ) + Nlocal ,

(10)

Term (a) is DKL pV = Φ(mV ). v∈V pv By linearity of expectation, term (b) splits into P E [log p (m )/q (m )] = N . p v v v v local v∈V The proof uses only Assumption 1 and linearity of expectation—no ordering, no Markov property—so the topology shapes pV without entering the statement, and independent messages give Φ = 0. In Example 1 the shortfall is Φ(m{u,v} ) = I(mu ; mv ), which reflects both the shared MOB counter and v’s persuasion. Three readings follow at once. Since Φ(mV ) ≥ 0 by Gibbs’ inequality, Njoint ≥ Nlocal , with equality iff the messages are mutually independent, so a perlink audit is optimistic and never pessimistic. The shortfall does not depend on the per-agent expectations: term (a) involves only the pv , never the qv , so Φ(mV ) survives any choice of qv , and no amount of care in modeling the senders recovers it. And because γ decreases with divergence while w increases with γ, the gap reaches the reported trust weights intact, rescaling the society’s confidence by a single factor. P Corollary 1 (Uniform SNR inflation). Let S ≜ v∈V Sv . The reported and joint-consistent cognitive SNRs γ rep = S/Nlocal and γ joint = S/Njoint satisfy γ rep Njoint Φ = =1+ ≜ β ≥ 1. γ joint Nlocal Nlocal

(12)

Proof: By (8), Sv depends only on what v transmitted, not on any reference law, so it is objective and receiver-computable yet intrinsic to the sender. The local evaluation sums these peragent signals. The joint evaluation reads the society’s asserted precision, which is block-diagonal because no agent P transmits cross-covariances; its trace is therefore the same v Sv = S. The two evaluations thus share the confidence S and differ only in their reference law, so S cancels and (12) is Theorem 1 divided by Nlocal . Since Φ(mV ) ≥ 0, β ≥ 1. This is what lets the shortfall be read as a power ratio without any commitment about how confidence is measured. Both evaluations share the same declared signal S, which cancels identically, so β is a dimensionless factor by which the audit over-reports the society’s cognitive SNR. Its decibel reading, 10 log10 β, is therefore an unqualified decibel, produced by the topology and the message correlations rather than by the agents’ calibration, and no recalibration can reduce it.

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IV. S CALING AND C ONSISTENCY T IME When a network scales, smaller local societies are glued into larger ones, as described in Section II-C. Gluing preserves locally certified states, and the consensus space H 0 = ker δ does not depend on the weights; so neither growth nor the trust inflation of Corollary 1 alters which agreements are valid. What both change is the rate of convergence—how fast agents agree, not whether they agree. To understand this speed limit, consider how the structure tracks variables. Because links compare only shared counters—comparing CP while ignoring UP, for instance—dimensions never mix, and the Laplacian decouples into independent reconciliation processes, one per dimension. Lemma 1 (The society splits by dimension). Let every restriction map be a coordinate projection onto the dimensions its link shares, and let Gc , the subgraph sharing dimension c, L be connected. The Laplacian decomposes as LF = c Lw Gc , where Lw is the graph Laplacian of G carrying the trust c Gc weights. The algebraic connectivity and the consistency time are therefore set entirely by the slowest dimension: λ2 (LF ) = min λ2 (Lw Gc ), c

τ (F) = max τ (Gc ). c

(13)

Proof: The Laplacian (5) computes differences across links with no cross-terms between dimensions. The spectrum of the direct sum is the union of the individual spectra, with unshared dimensions contributing only to the kernel. This creates a counter-intuitive dynamic: widely shared counters demand global consensus and become bottlenecks, while narrowly shared dimensions reconcile quickly across local links. Because Lemma 1 isolates each dimension into an ordinary graph Gc , classical graph theory dictates how topology governs the worst-case consistency time. Below, Br (v) denotes the radius-r ball around v, the agents within r hops of it, and a graph has polynomial growth d when its ball sizes scale as rd . Proposition 1 (Consistency time vs. topology). Let the consistency time (6) be evaluated for link weights w ≥ wmin > 0, on any sheaf F ′ over a subgraph of G. Topology dictates convergence: (i) Global scaling: On graphs with polynomial growth d (e.g., grids) and n agents, consistency slows as Ω(n2/d ) log(1/ε). Expander graphs, conversely, maintain a constant time Θ(1) log(1/ε) [14, Ch. 1–2]. (ii) Local scaling: Consistency within any ball Br (v) depends only on local geometry, independent of n: τ F|Br (v)

|Br (v)| diam(Br (v)) 1 ≤ log . 4 wmin ε

(14)

(iii) Universal bound: If the maximum degree ∆ is bounded, every radius-r ball reconciles in O(1) time with respect to n, whatever the global topology. Proof: By Lemma 1, it suffices to analyze each Gc , the slowest of which sets the overall speed. (i) On polynomialgrowth graphs, standard spectral theory gives λ2 = O(n−2/d ), which through (13) yields the Ω(n2/d ) bound. Expander graphs preserve a constant gap Θ(1) [14, Ch. 1–2], so the society is fast if every dimension expands. (ii) Restricting

TABLE I C ONSISTENCY TIME VERSUS TOPOLOGY AS n GROWS . F OR THESE NAMED FAMILIES THE ORDER IS TIGHT, SO THE Ω(n2/d ) OF P ROPOSITION 1 SHARPENS TO Θ(·). T HE LAST COLUMN IS THE EXPONENT OF n FITTED TO THE MEASURED CONSISTENCY TIME . Topology

d

λ2 (LG )

τ (F )

Measured

Path, Ring Grid Expander Complete

1 2 ∞ —

Θ(n−2 ) Θ(n−1 ) Θ(1) Θ(n)

Θ(n2 ) Θ(n) Θ(1) Θ(n−1 )

2.00, 1.99 0.98 −0.12 −1.03

each Gc to Br (v) gives a subgraph G′ with at most |Br (v)| vertices and diameter at most diam(Br (v)). Mohar’s lower bound on algebraic connectivity [15] states that for any graph G′ = (V ′ , E ′ ) with minimum edge weight wmin , λ2 ≥ 4wmin /(|V ′ | diam(G′ )). Inverting λ2 yields (14), which depends only on r and the maximum degree. (iii) With maximum P degree ∆, the Moore bound gives |Br (v)| ≤ 1 + ∆ k<r (∆ − 1)k [23], and diam(Br (v)) ≤ 2r, so (14) is O(1) in n. Table I summarizes this architectural reality: standard physical topologies such as paths and grids cannot achieve global consensus at scale, and only expanders or dense networks maintain a bounded global consistency time. Otherwise, architects must rely on (14), ensuring that well-connected local neighborhoods reconcile quickly, independent of the broader network. While Proposition 1 maps theoretical speeds, real-world auditing is imperfect. Applying Theorem 1 to the sub-society each link’s audit sees gives an inflation βe ≥ 1 per link e (Corollary 1), so the network is driven by distorted weights. The βe need not be equal across links, and Theorem 2 uses only βe ≥ 1, not their values. This creates a discrepancy between the reported consistency time τ rep , computed from the inflated weights, and the true consistency time τ joint , computed from the joint-consistent ones. Though the per-link audit cannot see this inflation, Theorem 2 proves the error is bounded, one-sided, and set by the worst-audited link. Theorem 2 (Consistency-time bias under inflated trust). For any sheaf F ′ and inflations βe ≥ 1, the ratio of the reported to the true consistency time is bounded by  −1 τ rep (F ′ ) max βe ≤ joint ′ ≤ 1. (15) e τ (F ) Proof: Because w(γ) is concave and increasing with w(0) = 0, the distorted weights w(βe γe ) are sandwiched as w(γe ) ≤ w(βe γe ) ≤ (maxe′ βe′ ) w(γe ). This ordering transfers to the Laplacians and hence to λ2 , and inverting yields (15). This theorem provides a predictable fail-safe. Because βe ≥ 1, the reported time τ rep is optimistic, but the error is tightly capped. (i) Lower bound: the network never synchronizes faster than reported, τ rep ≤ τ joint . (ii) Capped delay: errors do not compound; the true delay is capped by the single worstmeasured link, τ joint ≤ τ rep × maxe βe . (iii) Guaranteed budgets: multiplying the reported time by the maximum known inflation yields a safe upper bound, so if that bound meets the

1.2 1.0 0.8

15

13%

0.6

10%

10

0.4

5

0.2 2

3

4

agents in the society

5

0.0

4.5 4.0 3.5

0.60 dB

3.0 2.5 2.0 1.5

105

1.14 dB

reported, S/Nlocal joint-consistent, S/Njoint

0.48 dB

isolated

path

complete

Fig. 2. (a) Unseen divergence share Φ/Njoint across network topologies as a function of society size; the right axis carries the same measurement as the SNR penalty 10 log10 β. (b) Reported (S/Nlocal ) versus joint (S/Njoint ) cognitive SNR in dBcog ; their separation is that penalty at n = 5, and is independent of the reference (Corollary 1).

real-time requirement, the true system does too. This fail-safe complements the local scaling law: Corollary 2 (Locality improves the consistency time). Restricting the check to a ball Br (v) improves the error bound to (maxe⊆Br (v) βe )−1 , rendering the fidelity of the local consistency time immune to poorly audited links outside the ball. Locality thus offers a double advantage: local neighborhoods not only reconcile faster (Proposition 1), their reported consistency times are also more faithful (Corollary 2). A society can therefore guarantee reliable near-real-time local synchronization even if its distant periphery degrades. V. N UMERICAL R ESULTS We evaluate the framework in two regimes: elicited LLM agent societies, to test whether the trust shortfall of Theorem 1 is real, and simulated sheaves swept across topologies and sizes, to test the bounds of Proposition 1 and Theorem 2. A. Experimental Setup Our experiment runs five xApps with overlapping E2SMKPM observation masks (Fig. 1). All agents track a shared MOB counter (MM.HoExeFailCell), establishing a baseline correlation, and each is a locally hosted Mistral 7B model (temperature 0.6, a single sample per message). To separate interaction effects from that baseline, identical scenarios and seeds run across three topologies: (i) an isolated control without communication, where correlation stems solely from the shared counter; (ii) a path topology along a linear chain; and (iii) a complete topology with full peer context. Each agent reports its own assessment of the cell, using its neighbors’ messages purely as context rather than claims to restate, as in (1). The expectations qv are elicited from the same model given ov alone, with no neighbor messages. Generator and auditor therefore share a prior, which understates Nv : every number below is the optimistic end of what an operator auditing against an independent reference would measure. To measure asserted precision without altering live communication, each agent is prompted a second time with a fresh seed, and this second answer is never transmitted.

consistency time (ms)

23%

20

0

(b)

1.4

isolated (|E| = 0) path (|E| = 4) complete (|E| = 10)

cognitive SNR 10log10 (γ/γ0 ) (dBcog )

(a) 25

SNR penalty 10log10 β (dB)

divergence the audit never sees, Φ/Njoint (%)

6

104 near-real-time budget

103

n ∗ =13

n ∗ =172

102

path, global grid, global expander, global local, r = 3

101 101

agents n

102

Fig. 3. Consistency time versus society size (12 random sheaves per point; 10 ms per diffusion round). Solid lines denote global consistency times; dashed lines denote local radius-3 audits. The shaded band marks the 1 s near-realtime budget.

Pooled across scenarios, the squared difference between the 2 two responses gives σv,c , floored at two score points so that identical answers cannot generate infinite trust; the signal Sv from (8) is then the sum of the reciprocal variances. Because every quantity reported below is a ratio in which Sv cancels, none of them depends on this elicitation: it fixes the absolute SNRs of Fig. 2b, not their separation. B. The Composition Law Per-link audits miss joint divergence, creating a blind spot that expands with network scale and density (Fig. 2a). At n = 5, hidden divergence reaches 10%, 13% and 23% (β ≈ 1.12, 1.15, and 1.30) for the isolated, path, and complete societies. The 10% isolated baseline forms a structural floor from the shared counter. Communication expands the gap: a complete topology hides nearly double a path’s share, with 13 of its 23 points driven by peer persuasion rather than shared data. This shortfall distorts the trust layer by inflating reported confidence (Fig. 2b). Inter-agent context sharpens beliefs and raises both the reported and the joint cognitive SNR, but the reported one grows faster. Their separation is the penalty of Corollary 1, widening from 0.48 dB with isolated agents to 0.60 dB along a path and 1.14 dB in a complete society. Dense societies thus present a paradox: they report the highest confidence precisely where the audit harbors its largest blind spot. C. The Scaling Law Sweeping network size (n = 9 . . . 196) confirms that topology dictates real-time feasibility (Fig. 3), matching Table I: (i) path topologies scale quadratically, Θ(n2 ), exceeding the 1 s near-real-time budget at n∗ = 13 agents (0.5 s to > 200 s); (ii) grids scale linearly, Θ(n), breaching it at n∗ = 172 (60 ms to 1.1 s); and (iii) expanders maintain flat Θ(1) consistency times (100–400 ms), remaining within budget at every size. Local audits (dashed lines) bypass the global limits of sparse networks: a radius-3 audit converges in ≈ 300 ms on paths regardless of size, roughly 700× faster than the global consistency time at n = 196. On grids, local times plateau

7

multiplier needed τ joint /τ rep

practical strategies emerge to secure agentic control planes: (i) architecturally, by minimizing network diameter, deploying expanders, or relying on local audits; and (ii) informationally, by estimating unobserved message correlations directly from relayed traffic, without access to proprietary agent internals.

guarantee max βe e

6

whole society radius-r ball

5 4 3

R EFERENCES

2 1

1

2

3

4

guaranteed multiplier maxβe

5

6

e

Fig. 4. The multiplier τ joint /τ rep needed versus the one Theorem 2 guarantees, over 120 random sheaves on n = 100 agents, each certified globally and on a radius-3 ball.

near 200 ms. On expanders, local and global consistency times coincide, as Proposition 1 predicts: local shortcuts are unnecessary when global mixing is already fast. D. The Consistency-Time Law We evaluate Theorem 2 as an operational budgeting rule across 120 random sheaves, comparing the multiplier actually needed, τ joint /τ rep , against the maximum link inflation maxe βe (Fig. 4). The bound holds in every trial. Operators can therefore prevent timing violations by scaling the reported consistency time by maxe βe , computed from the message streams through Φ. Specifically, (i) trials land on the diagonal across scales, showing that the bound is tight; (ii) not knowing where the inflation sits costs only minor slack, under 1.5× (e.g., 3.0× needed against 3.7× guaranteed); and (iii) radius3 balls obey the same rule with the maximum taken over the links inside the ball (Corollary 2), none of their points lying above the diagonal. A neighborhood’s consistency time can thus be budgeted from the inflation of its own links alone, whatever the audit quality elsewhere in the society. VI. C ONCLUSION Link-by-link trust evaluation keeps an O-RAN RIC within its tight control loop but obscures how agent conclusions depend on one another. This blind spot is exact: the joint divergence exceeds the reported local sum by the total correlation of the transmitted messages. In agent societies this creates a trust paradox: as communication increases, reported confidence rises while unobserved dependence hides up to 23% of the joint divergence in dense networks. Modeling the control plane as a cellular sheaf shows that this mis-measured audit still certifies valid consensus states, sacrificing only convergence speed. Reconciliation decomposes by metric counter, making consensus as slow as the worst-connected dimension, and consistency time grows quadratically with network diameter: path topologies breach a 1 s near-real-time budget at n∗ = 13 agents and grids at n∗ = 172, whereas radius3 local audits converge in ≈ 300 ms regardless of network size. Multiplying the reported consistency time by the worstlink inflation maxe βe provides a safe operational bound. Two

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