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.
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
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
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
1.4 Iteration Coherence#
Stability of operator iteration.
Coherent when:
- maps and flows remain valid
- iteration does not introduce instability
- operator composition remains deterministic
1.5 Geometric Coherence#
Compatibility with state‑space geometry.
Coherent when:
- trajectories respect geometric constraints
- attractors embed correctly
- divergence aligns with geometry
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.
C1 — Weak Coherence#
- partial divergence stability
- fragile attractor structure
- iteration unstable
Chaos cannot sustain.
C2 — Moderate Coherence#
- bounded divergence
- stable iteration
- attractor formation begins
Chaos emerging.
C3 — Strong Coherence#
- stable fractal attractors
- deterministic divergence
- multi‑scale structure
- geometry compatible
Full chaotic behavior supported.
C4 — Perfect Coherence (Ideal)#
- perfect attractor stability
- perfect divergence structure
- perfect iteration stability
C4 is theoretical; real systems approach C3.
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.
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.
R2 — Transitional / Moderate‑Sensitivity#
- bifurcations
- emerging fractal structure
- partial coherence decay
Coherence dominated by attractor formation.
R3 — Fully Chaotic / High‑Sensitivity#
- exponential divergence
- fractal attractors
- multi‑scale structure
Coherence dominated by divergence structure + attractor stability.
5. Coherence Evaluation Procedure#
To evaluate coherence:
- Validate sensitivity structure
- Validate divergence behavior
- Validate attractor geometry
- Validate iteration stability
- Validate geometric compatibility
- Validate regime alignment
If any step fails → classify collapse mode.
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.