Coherence Map — Information Theory

TriadicFrameworks /docs/theories/information_theory/coherence_map.md#

Information Theory in TriadicFrameworks defines coherence as distinction stability under operator action.
Coherence is structural, not probabilistic.
Signals are operators, not messages.
Distinctions must remain identifiable, non‑degenerate, and operator‑consistent across regimes.

This file defines the coherence dimensions, coherence levels, collapse modes, and regime behavior for Information Theory.

1. Coherence Dimensions#

Information Theory uses five structural coherence dimensions:

1.1 Distinction Coherence#

Stability of distinctions as structural units.

A distinction is coherent when:

• it remains identifiable
• it does not degenerate
• it preserves its invariants

1.2 Operator Coherence#

Stability of distinctions under operator action.

An operator is coherent when:

• it preserves distinction identity
• it does not introduce ambiguity
• it maintains dimensional consistency

1.3 Adjacency Coherence#

Stability of structural distances between distinctions.

Adjacency is coherent when:

• distances remain consistent
• no adjacency inversion occurs
• no collapse of structural neighborhoods

1.4 Dimensional Coherence#

Stability of dimensional profiles.

Dimensional coherence holds when:

• dimensional profiles remain valid
• no dimensional drift occurs
• transforms preserve dimensional identity

1.5 Regime Coherence#

Stability of distinctions across R0 → R3 transitions.

Regime coherence holds when:

• transitions preserve identity
• transitions preserve coherence
• transitions do not introduce collapse

2. Coherence Levels (C0 → C4)#

Coherence is evaluated on a five‑level structural scale:

C0 — Incoherent#

• distinctions unstable
• operators undefined
• dimensional profile invalid

System cannot support information.

C1 — Weak Coherence#

• distinctions exist but unstable
• adjacency inconsistent
• operator action unreliable

System supports only primitive structure.

C2 — Moderate Coherence#

• distinctions stable
• operators valid
• adjacency mostly consistent

System supports basic information structure.

C3 — Strong Coherence#

• distinctions stable under operators
• adjacency consistent
• dimensional profiles valid
• regime transitions stable

System supports full distinction‑based information.

C4 — Perfect Coherence (Ideal)#

• distinctions fully stable
• operators fully stable
• dimensional and adjacency coherence perfect
• regime transitions lossless

C4 is theoretical; real systems approach C3.

3. Collapse Modes (Structural)#

Collapse occurs when distinctions fail structurally.

C1 — Distinction Ambiguity#

Distinctions lose identity.

C2 — Dimensional Inconsistency#

Dimensional profiles become invalid.

C3 — Operator Instability#

Operators fail to preserve distinctions.

C4 — Coherence Failure#

System‑wide structural collapse.

Collapse is structural, not probabilistic.

4. Regime Behavior (R0 → R3)#

Coherence behaves differently across RTT regimes:

R0 — Pre‑Distinction#

• no stable distinctions
• coherence undefined

R1 — Distinction Stability#

• distinctions stable
• minimal operator coherence

R2 — Operator Geometry#

• operator coherence primary
• adjacency coherence active

R3 — Dimensional Operators#

• dimensional coherence active
• regime coherence required

5. Coherence Evaluation Procedure#

To evaluate coherence:

1. Validate distinction identity
2. Validate operator stability
3. Validate adjacency consistency
4. Validate dimensional profile
5. Validate regime compatibility

If any step fails → classify collapse mode.

6. Summary#

Information Theory coherence is:

• distinction‑first
• operator‑driven
• structural
• regime‑aware
• zero drift

Information = structured distinction.
Coherence = distinction stability.
Signals = operators acting on distinction spaces.