Conflict is often treated as a defect to be hidden by better prompting, stronger retrieval, or post-hoc explanation. Part I argued that conflict is a signal of structural honesty. This section formalizes that claim. The objective is not only to detect inconsistency, but to measure and route it: what is incompatible, where it lives, how severe it is, which actors contributed to it, and what action consequences follow.
5.1 Conflict semantics
A conflict is not merely syntactic disagreement. It is an incompatibility relation induced by domain semantics, scope, and policy: where iff assertions and are jointly inadmissible under the active policy and scope constraints.
Many apparent contradictions are not contradictions once scope is respected, while some semantically incompatible claims may look lexically similar. Governance systems therefore require conflict semantics, not string mismatch tests.
5.2 Conflict graph and severity
Given a governed snapshot , define the conflict graph where are active assertions, , and is a severity weight.
Severity may be defined as a composite function of action relevance, policy criticality, evidentiary confidence asymmetry, temporal proximity, and dependency centrality. The weighting function is implementation-specific. We require only that be nonnegative and measurable.
5.3 Spectral decomposition
Let denote the weighted adjacency matrix of . Spectral structure provides a compact characterization of conflict topology: connected components identify isolated contradiction regions, principal eigenvectors identify conflict concentration and propagation pathways, and spectral gap provides a proxy for cluster separation.
Proposition 1 - Conflict decomposability
If the conflict graph has connected components , then (i) total conflict energy decomposes as , and (ii) resolving all conflicts within component does not change the conflict status of assertions in any other component ().
Proof sketch. Conflict edges exist only within components by definition. Energy is a sum over edges, which partitions across components. Conflict status depends only on the existence and severity of edges incident to an assertion; removing edges within affects only assertions in . ∎
The operational consequence is that conflict resolution parallelizes naturally across independent contradiction regions. A governance system should exploit this decomposition: resolve components independently, prioritize the component with highest energy concentration, and avoid serializing resolution work across unrelated disputes.
5.4 Conflict energy
A normalized variant for cross-snapshot comparisons: where avoids division by zero.
Proposition 2 - Monotonicity of conflict energy under assertion addition
Under fixed policy and fixed severity function , adding assertions to a governed snapshot cannot decrease total conflict energy:
Proof sketch. New assertions may introduce new conflict edges but cannot remove existing ones under fixed policy (by Theorem 4). Each new edge contributes non-negative weight. Therefore total conflict energy is non-decreasing. ∎
This is the quantitative counterpart to Theorem 4's set-inclusion result. Theorem 4 says conflict awareness cannot decrease. Proposition 2 says conflict burden cannot decrease. Together they formalize why governance systems must treat conflict resolution as an active process, not a passive hope that new information will dilute contradictions.
5.5 Multi-agent fusion and the trust tensor
In multi-agent settings, conflicts are shaped not only by claims but by the reliability and interaction structure of the claiming agents.
Let agents be indexed by . We define a trust state tensor that conditions contribution quality over agent identity, task or domain class, scope regime, evidence modality, and time:
representing estimated contribution reliability of agent in domain , scope regime , at time . Fusion then becomes a trust problem, not a simple average:
where each support record contributes claim confidence , agent-conditioned trust, and evidence . The choice of fusion operator is open; the key requirement is that it remain auditable and policy-compatible.
5.6 The trust formation game
To formalize incentives in multi-agent systems, consider a stylized repeated game in which agents propose state updates and action proposals. Agent utility under governance:
where is task reward, penalizes unresolved conflict, penalizes blocked action proposals, penalizes provenance gaps, and rewards successful disambiguation.
This game differs from generic coordination games because action feasibility is endogenously altered by the governance layer.
5.7 Equilibrium under trust constraints
Conjecture 1 - Governance-induced equilibrium shift
Let be the multi-agent task game without state-integrity gating, and the governed variant. Under the following conditions: (i) audit outcomes are observable to all agents, (ii) penalty coefficients are sufficiently large relative to task reward variance, (iii) the governance layer satisfies the monotonicity properties of Theorems 4 and 5, any stationary equilibrium of that maximizes long-run expected reward weakly reduces expected unresolved conflict energy relative to corresponding equilibria in , subject to bounded task-performance degradation.
This is stated as a conjecture because a full proof requires specifying the strategy space, information structure, and equilibrium concept more precisely than this paper attempts. Its practical importance is immediate even without proof: a trust layer can reshape agent behavior toward evidentiary discipline even when base models remain probabilistic and imperfect. Trust constraints do not merely limit agents. They change what rational behavior looks like.