Conjecture 6.8: Group Structure Discovery

VALIDATED The interaction matrix discovers coalition topology: aligned objectives cluster, competing objectives separate.

Statement

When objectives naturally form coalitions (subsets with aligned gradients), the interaction matrix \( A_{ij} \) will exhibit block structure that reveals these coalitions. Within-coalition entries will have \( \alpha \approx 0 \) (aligned) and between-coalition entries will have \( \alpha \approx \pi/4 \) (partially competing).

Status: Validated

The interaction matrix reliably discovers group structure:

Within-group: \( \alpha = 0 \) — gradients perfectly aligned, no projection needed
Between-group: \( \alpha \approx 0.785 \) radians (\( \approx 45° \)) — the cosine-scaled projection applies moderate correction

Evidence Summary

The experiment exp_symmetry_breaking.sx creates systems with known coalition structure (e.g., 6 objectives in 2 groups of 3) and verifies that the interaction matrix discovers the partition:

Block-diagonal structure emerges within 5 adaptation cycles
Within-block entries converge to \( \alpha \approx 0 \) (mean: 0.003)
Between-block entries converge to \( \alpha \approx 0.785 \) (mean: 0.782)
The between-group angle of \( \pi/4 \) is notable — it represents the natural equilibrium where the cosine-scaled projection removes exactly half the conflicting component

The \( \pi/4 \) angle is not imposed by the algorithm; it emerges from the dynamics. This suggests that composed adaptive systems self-organise to a state where inter-group conflicts are at a natural balance point.

Relevant Experiments

exp_symmetry_breaking.sx — coalition discovery and symmetry breaking
exp_interaction_matrix.sx — interaction matrix dynamics
exp_structure_discovery.sx — general topology recovery

What This Means

Group structure discovery has direct practical value: in complex multi-objective systems, knowing which objectives are aligned and which compete allows targeted resource allocation. The emergence of the \( \pi/4 \) between-group angle is theoretically interesting — it suggests a universal equilibrium angle for inter-coalition conflict, analogous to the \( I = -1/2 \) equilibrium of the I-Ratio Theorem (Theorem 13).