Information Geometry of Nested Frequency Hierarchies: Why Biological Systems Exhibit High-Dimensional Coherence
What's this about?
Nested frequency hierarchies are the geometric foundation of high-dimensional coherence. Biological systems from neural circuits to circadian rhythms exhibit oscillations at multiple timescales with systematic phase-amplitude coupling. This paper proves why this architecture creates high-dimensional dynamics.
The key insight: each frequency band contributes a phase degree of freedom. Nesting couples these into a product manifold T^{K-1} whose dimension equals the number of bands minus one.
Three theorems:
- Product Structure: K nested frequency bands create a (K-1)-dimensional phase torus
- Coupling Activates Rank: Inter-band phase-amplitude coupling increases Fisher rank by activating off-diagonal information
- Clock Collapse: Global clock synchronization collapses the product manifold to dimension 1
The Fisher rank gap between biological systems (≥K-1) and clocked digital systems (≤1) provides a geometric explanation for persistent differences between biological and artificial intelligence.
Key findings
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K nested frequency bands → (K-1)-dimensional phase manifold
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Phase-amplitude coupling activates off-diagonal Fisher information
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Clock synchronization reduces Fisher rank to at most 1
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Dimensional gap: biological (≥K-1) vs clocked digital (≤1)
Citation
Todd, I. (2026). Information Geometry of Nested Frequency Hierarchies: Why Biological Systems Exhibit High-Dimensional Coherence. Information Geometry (in preparation).
Workflow: Claude Code with Opus 4.5 (Anthropic) for drafting. Author reviewed all content and takes full responsibility.