Modal Logic for Distributed Trust

Layered trust models for multi-agent LLM systems remain largely conceptual: they name which dimensions of trust matter but not how layers combine, how their importance is set at runtime, or how trust should govern agent actions. This gap matters because higher inter-agent trust raises task success while also enlarging exposure to exploitation, a tension formalized as the Trust-Vulnerability Paradox. We make a five-layer trust stack operational through three contributions. First, a cross-layer synergy operator propagates prerequisite-layer deficits into de- pendent layers, feeding a generalized-mean composite trust that recovers the weakest-link rule as a limiting case, with provable bounds. Second, per-layer importance weights are grounded in observed failures via a no-regret online estimator that tracks which layer is currently most responsible for harm. Third, Trust- Gated Capability Control issues short-lived, revocable capability grants only when composite trust and the relevant prerequisite layers clear capability-specific thresholds. We prove this mechanism breaks the paradox: a stealthy compromise that inflates behavioral trust while degrading a prerequisite layer cannot escalate privilege, and give a closed-form, provably conservative trust fixed point with a bounded-latency revocation guarantee. A numerical study with a sleeper adversary confirms the predictions: composite-trust bounds hold across 20,000 random draws, the analytic fixed point matches simulation within 0.007, and a compromised prerequisite layer triggers automatic revocation within a few interactions.

As language-model agents increasingly work in teams, each agent must decide how much to trust its teammates. Yet we lack a standard way to measure trust between AI agents. We propose a behavioral measure based on costly verification. In a cooperative survival game, checking a teammate's work consumes resources, while trusting a wrong answer can be fatal. Relative to a memoryless version of the same model, reduced verification provides an observable measure of trust. Using this framework, we study trust formation, breakage, and recovery across six frontier model snapshots. When paired with a consistently reliable teammate, four snapshots (Claude Opus 4.6, Claude Sonnet 4.6, GPT-5.1, and Gemini 3.1 Pro) reduce verification by roughly 60-85%, whereas two smaller snapshots show little or no such adjustment. Failures reverse this discount, but models differ in how they respond. Some concentrate renewed scrutiny on the culprit, while others become more cautious toward the entire team. Recovery is slower than formation, and clustered failures sustain suspicion far longer than the same number of failures spread apart. These differences have practical consequences. Models that form trust verify less, decide more quickly, and achieve higher payoffs in our environment. By contrast, persistent over-verification is associated with indecision rather than safety. Our results show that trust dispositions can be measured before deployment and suggest that calibration, rather than maximal suspicion, should be the central concern in the governance of multi-agent AI systems.

Multi-agent LLM systems exchange messages before acting through tools, but message delivery, signing authority, and tool-side enforcement are separate trust boundaries. We represent them as communication, authorization, and execution structures, and use the authority cut to characterize the smallest independently compromised signer coalition that can construct a policy-valid false-claim proof under stated assumptions. Controlled paired hidden-state studies reveal four distinctions. First, no fixed-message routing contrast across four models meets the prespecified large-effect criterion; when relays rewrite messages, executed false-state action rates vary by model and route configuration. Second, presented source-identity cues do not uniformly increase protected-action selection across three models. Third, on invalid-proof cases, Nemotron and Llama request protected actions in 52.1% and 54.2% of trials, respectively, yet a deterministic tool gate commits none: request-level and commitment-level safety readouts diverge. Finally, a separate single-model authorization check shows that a false claim with a policy-valid compromised-signer proof can pass the gate, since proof validity is not factual truth. These findings motivate evaluating delivered information, qualified authorization, model proposals, and committed actions separately.