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Conjecture 6.4: Belief Chain Discovery

VALIDATED The interaction matrix can recover causal chain structure among belief agents from adaptation dynamics alone.

Statement

When belief agents are arranged in a causal chain (agent \( i \) influences agent \( i+1 \)), the interaction matrix \( A_{ij} \) computed from gradient inner products will exhibit a band structure that reveals the chain topology — even without prior knowledge of the arrangement.

Status: Validated

The interaction matrix partially discovers the chain structure from data. Adjacent agents show strong interaction (\( |A_{ij}| \) large for \( |i-j| = 1 \)) while distant agents show weak interaction (\( |A_{ij}| \approx 0 \) for \( |i-j| > 2 \)).

Evidence Summary

The experiment exp_belief_cascade.sx sets up chains of 5–8 belief agents with known causal ordering and computes the interaction matrix after convergence:

  • Adjacent-agent interactions are 5–10x stronger than non-adjacent
  • The band structure of \( A \) correctly identifies nearest-neighbour relationships
  • Discovery is "partial" because second-order effects (agent \( i \) to \( i+2 \)) are present but weaker, sometimes below noise

This validates the conjecture with the qualifier that the method discovers the topology partially — nearest-neighbour structure is reliable, but longer-range dependencies require more data or higher-order analysis.

Relevant Experiments

  • exp_belief_cascade.sx — chain topology discovery from gradient interactions
  • exp_interaction_matrix.sx — interaction matrix convergence dynamics
  • exp_structure_discovery.sx — general topology recovery from gradient structure

What This Means

The interaction matrix is not just a theoretical tool for proving convergence — it is a practical instrument for structure discovery. Given a set of interacting adaptive agents, the matrix reveals which agents influence which, without requiring prior knowledge of the topology. This has implications for debugging complex multi-agent systems and for understanding emergent structure in neural architectures.