# Coherence Map — Chaos Theory  
### TriadicFrameworks /docs/theories/chaos_theory/coherence_map.md

Chaos Theory in TriadicFrameworks is a **structural sensitivity theory**,
not a randomness theory and not a pop‑science “butterfly effect”
narrative.

Chaos = **deterministic sensitivity to operator iteration**.  
Attractors = **coherence surfaces**, not metaphors.  
Unpredictability = **coherence decay**, not randomness.

This file defines how coherence is evaluated across operators,
trajectories, attractors, geometry, and RTT regimes.

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# 1. Coherence Dimensions

Chaos Theory uses **five structural coherence dimensions**:

## 1.1 Sensitivity Coherence  
Stability of sensitivity under iteration.

Coherent when:

- divergence is bounded  
- sensitivity amplification is structural  
- no randomness is introduced  

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## 1.2 Divergence Coherence  
Stability of trajectory separation.

Coherent when:

- divergence follows deterministic structure  
- exponential divergence is bounded by attractor geometry  
- divergence does not collapse into noise  

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## 1.3 Attractor Coherence  
Stability of attractor geometry.

Coherent when:

- attractor structure is consistent  
- fractal geometry is stable  
- trajectories remain bounded  
- no topological collapse occurs  

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## 1.4 Iteration Coherence  
Stability of operator iteration.

Coherent when:

- maps and flows remain valid  
- iteration does not introduce instability  
- operator composition remains deterministic  

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## 1.5 Geometric Coherence  
Compatibility with state‑space geometry.

Coherent when:

- trajectories respect geometric constraints  
- attractors embed correctly  
- divergence aligns with geometry  

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# 2. Coherence Levels (C0 → C4)

Coherence is evaluated on a **five‑level structural scale**:

## **C0 — Incoherent**  
- unbounded divergence  
- invalid attractor structure  
- operator instability  
- geometry incompatible  

System cannot support chaotic behavior.

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## **C1 — Weak Coherence**  
- partial divergence stability  
- fragile attractor structure  
- iteration unstable  

Chaos cannot sustain.

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## **C2 — Moderate Coherence**  
- bounded divergence  
- stable iteration  
- attractor formation begins  

Chaos emerging.

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## **C3 — Strong Coherence**  
- stable fractal attractors  
- deterministic divergence  
- multi‑scale structure  
- geometry compatible  

Full chaotic behavior supported.

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## **C4 — Perfect Coherence (Ideal)**  
- perfect attractor stability  
- perfect divergence structure  
- perfect iteration stability  

C4 is theoretical; real systems approach C3.

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# 3. Collapse Modes (CH1 → CH5)

Collapse occurs when coherence fails structurally.

## **CH1 — Operator Collapse**  
Invalid map/flow.

## **CH2 — Divergence Collapse**  
Unbounded or undefined divergence.

## **CH3 — Coherence Collapse**  
Iteration instability.

## **CH4 — Parameter Collapse**  
Invalid parameter region.

## **CH5 — Geometry Collapse**  
State‑space incompatibility.

Collapse is structural, not random.

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# 4. Regime Behavior (R1 → R3)

Coherence behaves differently across RTT regimes:

## **R1 — Stable / Low‑Sensitivity**  
- bounded divergence  
- stable iteration  
- simple attractors  

Coherence dominated by iteration stability.

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## **R2 — Transitional / Moderate‑Sensitivity**  
- bifurcations  
- emerging fractal structure  
- partial coherence decay  

Coherence dominated by attractor formation.

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## **R3 — Fully Chaotic / High‑Sensitivity**  
- exponential divergence  
- fractal attractors  
- multi‑scale structure  

Coherence dominated by divergence structure + attractor stability.

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# 5. Coherence Evaluation Procedure

To evaluate coherence:

1. Validate sensitivity structure  
2. Validate divergence behavior  
3. Validate attractor geometry  
4. Validate iteration stability  
5. Validate geometric compatibility  
6. Validate regime alignment  

If any step fails → classify collapse mode.

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# 6. Summary

Chaos Theory coherence is:

- **structural**  
- **deterministic**  
- **operator‑driven**  
- **multi‑scale**  
- **geometry‑embedded**  
- **regime‑aware**  
- **zero drift**

Chaos = **deterministic structural sensitivity**.  
Attractors = **coherence surfaces**.  
Dynamics = **operator‑driven iteration**.