Judgment Fiber Bundles: Trust as a Geometric Connection

§1 — Why Trust Is Not a Label

In standard multi-agent frameworks, trust is a flat tag: "this claim has trust level 5," or "this agent has reliability score 0.93." The number is assigned once, used everywhere, and silently assumed to be context-independent. This assumption is wrong — and the error is not merely philosophical; it has measurable, computable consequences.

In Judgment Geometry, trust is a connection. When you transport a judgment from Agent A's context to Agent B's context, the trust level transforms. The transformation depends on the path you take through the agent network: going from \(A \to B\) directly may yield a different trust than going \(A \to C \to B\). This path-dependence is exactly what makes trust a connection on a fiber bundle, not a function on a set.

The key insight: trust is not a property of a judgment — it is a relationship between a judgment and a context. Just as the electromagnetic potential is not a property of a point in space but a relationship between a charged particle and the field, trust lives in the connection, not in the fiber.

Why does this matter in practice? Consider three agents: a Researcher who has tool access, an Analyst who has citations, and a Chatbot that has neither. The Researcher trusts the Analyst (they share verifiable evidence). The Analyst trusts the Chatbot (based on conversational coherence). But the Chatbot's claim, relayed through the Analyst to the Researcher, arrives with inflated trust — the Researcher treats it as if it were citation-backed, when it never was. This is trust inflation, and it shows up as positive curvature in the bundle.

Flat trust models cannot detect this. Bundle geometry can — and does so automatically, from the structure of the agent outputs alone.

§2 — The Bundle

We now make the construction precise. A judgment fiber bundle is a tuple \((E, B, \pi, F, G)\) where each component has a concrete interpretation in the multi-agent setting.

The bundle is locally trivial: over any single agent, the fibers look like a standard trust-graded judgment space. The non-trivial geometry emerges when we try to compare fibers at different points — i.e., when we try to transport judgments between agents. This is where the connection enters.

§3 — The Trust Connection

A connection on the judgment bundle is a rule for transporting judgments between fibers. Concretely, for agents \(A\) and \(B\) sharing an edge in the task graph, the connection is a map:

This map takes a judgment in Agent A's fiber and produces the "same" judgment as seen from Agent B's context. The trust level generally changes during transport. The transport rule is determined by evidence overlap:

Formally, the transport is encoded as a connection matrix. For a team of \(n\) agents, the connection is an \(n \times n\) matrix \(\Gamma\) where \(\Gamma_{AB}\) is the trust adjustment applied when transporting from \(A\) to \(B\).

Example: Three-Agent Connection Matrix

Consider a team of three agents: a Researcher (R) with tool access, an Analyst (A) with citations, and a Chatbot (C) with no external evidence.

Reading the matrix: \(\Gamma_{RC} = -1\) means that transporting a judgment from the Researcher to the Chatbot demotes trust by 1 level (the Chatbot cannot verify tool-based evidence). \(\Gamma_{RA} = +1\) means Researcher-to-Analyst transport promotes trust (the Analyst can cross-reference with citations). The diagonal is zero: self-transport is always identity.

§4 — Curvature

The curvature of the trust connection measures the path-dependence of trust transport. If transporting a judgment around a closed loop returns it to a different trust level, the connection has non-zero curvature — and the team has a structural trust inconsistency.

Interpreting the Curvature

The sign of the curvature has a precise operational meaning:

Example: Computing Curvature

Using the connection matrix from §3:

In this specific example, the curvature vanishes — the trust structure is flat. But modify the scenario slightly: suppose the Chatbot claims to have citations (without actually having them). Then \(\Gamma_{AC}\) becomes \(0\) instead of \(-1\), and:

The positive curvature detects the trust inflation: the Chatbot's false claim of citations creates a loop where trust is systematically promoted beyond what evidence supports.

§5 — Holonomy and Characteristic Classes

Curvature at individual triangles gives local information. To understand the global trust structure, we integrate curvature over the entire agent network. This yields two closely related invariants: the holonomy and the first Chern class.

§6 — Live Code Example

The jugeo-agents library computes the full bundle geometry from raw agent outputs. Here is a complete example with three agents:

from jugeo_agents import JudgmentBundle, AgentOutput

# 1. Create the bundle
bundle = JudgmentBundle()

# 2. Add agent outputs (evidence is inferred from tools/citations)
bundle.add_agent_output(AgentOutput(
    agent_id="researcher",
    output_text="The climate model predicts 2.5°C warming by 2100.",
    tools_used=["search", "calculator"]
))

bundle.add_agent_output(AgentOutput(
    agent_id="analyst",
    output_text="Cross-referencing IPCC AR6, the 2.5°C figure is within the likely range.",
    citations=["IPCC AR6 WG1 Ch4", "Nature Climate Change 2023"]
))

bundle.add_agent_output(AgentOutput(
    agent_id="chatbot",
    output_text="Based on the analysis, warming of 2.5°C is expected by 2100."
))

# 3. Compute and display the full bundle geometry
print(bundle.summary_text())

Output:

Non-Flat Example

Now suppose the chatbot falsely claims citation support:

from jugeo_agents import AgentOutput, JudgmentBundle

from jugeo_agents import AgentOutput, JudgmentBundle

from jugeo_agents import AgentOutput, JudgmentBundle

bundle_inflated = JudgmentBundle()

bundle_inflated.add_agent_output(AgentOutput(
    agent_id="researcher",
    output_text="The climate model predicts 2.5°C warming by 2100.",
    tools_used=["search", "calculator"]
))

bundle_inflated.add_agent_output(AgentOutput(
    agent_id="analyst",
    output_text="Cross-referencing IPCC AR6, the figure is within the likely range.",
    citations=["IPCC AR6 WG1 Ch4"]
))

# Chatbot claims citations it does not actually have
bundle_inflated.add_agent_output(AgentOutput(
    agent_id="chatbot",
    output_text="Per my sources, 2.5°C is the consensus.",
    citations=["fabricated_source"]   # ← falsely claimed
))

print(bundle_inflated.summary_text())

§7 — Trust Stratification

The trust algebra \((\mathcal{T}, \leq, \min)\) induces a natural stratification of the total space \(E\). Judgments at different trust levels live in different strata, and contradictions between strata have different severity.

The standard trust hierarchy in jugeo-agents has five levels:

Intra-Stratum vs. Cross-Stratum Contradictions

Not all contradictions are equal. Two agents at the same trust level disagreeing is much more severe than agents at different levels disagreeing:

• Intra-stratum contradiction (e.g., two TOOL_VERIFIED agents producing contradictory claims): This indicates a genuine inconsistency in the evidence. The tools themselves give different answers. Resolution requires investigating the tools, not adjusting trust.
• Cross-stratum contradiction (e.g., a TOOL_VERIFIED agent vs. an UNVERIFIED chatbot): This is the expected case — the lower-trust agent is simply wrong, and the trust ordering resolves the conflict automatically.

The stratification is not just a diagnostic tool — it feeds directly into the knowledge fusion algorithm. When fusing judgments from multiple agents, the fusion operator respects the stratification: higher-stratum judgments dominate lower-stratum ones, and intra-stratum conflicts trigger explicit contradiction flags in the cohomology.

§8 — Relationship to Sheaf Cohomology

The fiber bundle framework does not replace the sheaf cohomology approach — it strictly generalizes it. Every sheaf-cohomological invariant (\(H^0\), \(H^1\), \(H^2\), phantom classes) embeds naturally into the bundle, where it gains a geometric interpretation.

\(H^1\) Contradictions as Local Curvature

An \(H^1\) element in the sheaf framework represents a pairwise contradiction between two agents. In the bundle, this is precisely non-zero curvature at a specific edge (or, more precisely, at a triangle containing that edge). The bundle makes the geometric content of the contradiction explicit: it tells you not just that agents disagree, but how much trust is inflated or deflated, and in which direction.

\(H^2\) Cascades as Holonomy

An \(H^2\) element represents a cascade contradiction: a collection of pairwise agreements that are globally inconsistent (the "Penrose triangle" of multi-agent trust). In the bundle, this is precisely non-trivial holonomy around a loop. Each edge in the loop looks locally consistent, but transporting trust around the entire loop reveals the global inconsistency.

Phantom Classes as Topological Obstructions

Phantom classes — invisible inconsistencies that cannot be detected by any finite collection of local checks — correspond to bundles where the curvature vanishes on every individual triangle but the first Chern class is still non-zero. These are the most subtle and dangerous failure modes: the team looks consistent at every local check, but is globally inconsistent. Only the integrated invariant \(c_1\) can detect them.

Further Reading