Cascades and Critical Transitions
A Formal Mapping Between Cascade Dynamics, Phase Transitions, and Clarus Five‑Node Geometry
Cascades are not metaphors. They are phase transitions on networks, and once you recognise this, renormalization group becomes the natural language for lock‑in, recoverability collapse, and interaction‑order geometry.
The document shows how branching processes, percolation, threshold models, and self‑organized criticality all share the same critical structure. It explains how coarse‑graining a cascade produces RG flow, fixed points, and hysteresis, revealing why small parameter shifts can trigger sudden regime changes.
It then maps Clarus’s five‑node variables directly onto the control parameters and dynamical responses that determine distance to criticality. This makes recoverability measurable through spectral gaps, susceptibility, correlation length proxies, and critical slowing down. The document provides the mathematical backbone that turns Clarus cascade language into a defensible, physics‑aligned framework.
Index
The document shows how branching processes, percolation, threshold models, and self‑organized criticality all share the same critical structure. It explains how coarse‑graining a cascade produces RG flow, fixed points, and hysteresis, revealing why small parameter shifts can trigger sudden regime changes.
It then maps Clarus’s five‑node variables directly onto the control parameters and dynamical responses that determine distance to criticality. This makes recoverability measurable through spectral gaps, susceptibility, correlation length proxies, and critical slowing down. The document provides the mathematical backbone that turns Clarus cascade language into a defensible, physics‑aligned framework.
Index
1. Cascades as Phase Transitions
Universal structure of cascades and the critical boundary where propagation becomes self‑sustaining.
2. Canonical Models
Branching processes, percolation, threshold cascades, and self‑organized criticality as formal phase transitions.
3. Renormalization Group View
Coarse‑graining cascades, parameter flow, fixed points, and the basin structure behind lock‑in.
4. Distance‑to‑Criticality Indicators
Susceptibility, correlation length, critical slowing down, and power‑law cascade signatures.
5. Clarus Five‑Node Mapping
Pressure, buffer, lag, coupling tightness, and cascade state as decomposed control parameters.
6. Composite Control Parameter
Amplification divided by dissipation as the structural form of the transition boundary.
7. Lock‑In and Hysteresis
Irreversibility windows, basin capture, and recoverability collapse.
8. K and E as Mathematical Quantities
Coherence as spectral gap; uncertainty distribution as variance spread across nodes.
9. Practical Indicators
Largest eigenvalue, return‑time, susceptibility, heavy‑tail emergence, and finite‑size scaling.
10. One‑Paragraph Synthesis
Cascades as critical phenomena, RG explaining lock‑in, and Clarus variables decomposing the critical manifold.