Mathematical Foundations of Circulatory Epistemology
The Simplex
Path Divergence on the Information Geometry
KL Divergence: D_KL(P||Q) = 0.727 | D_KL(Q||P) = 0.587 | Ratio = 0.81 Paths: Gold line shows e-geodesic (exponential family), steel line shows m-geodesic (mixture family). The spacing reveals asymmetry in information space.
The Sparse Loop
Intervention Efficiency and Convergence
Final Distances: Autonomous = 0.142 | Sparse (5 interventions) = 0.038 | Dense (20 interventions) = 0.014 The Pattern: Without engagement, the system drifts. With sparse loops, it corrects. With continuous loops, it stabilizes. Each intervention is a breath.
The Learning Rate
Timing in the Loop (λ = Information Fraction)
0.50
Interpretation: At λ = 0.50 (balanced timing), all strategies converge. Front-loading fails early; back-loading saves late. The loop breathes best when timing is distributed.
The Freezing Lemma
Variance Decomposition as λ Increases
Key Transition: At λ ≈ 0.85, between-group variance dominates. System coherence increases from individual exploration to collective alignment. Reduction: Total variance drops 16,000× from λ=0 to λ=1. When timing becomes perfect, noise becomes silence.