# Operators — Information Theory  
### TriadicFrameworks /docs/theories/information_theory/operators.md

Information Theory in TriadicFrameworks is a **distinction‑first
coherence grammar**. Operators act on **distinction spaces**, not on
probabilities, messages, or semantic content. Signals are operators;
coherence is distinction stability; information is structured
distinction.

This file defines the canonical operators for Information Theory across
R0 → R3.

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# Operator List

The core operators are:

- **𝓓** — distinction operator  
- **𝓢** — signal operator  
- **𝓒** — coherence operator  
- **𝓐** — adjacency operator  
- **𝓣** — transform operator  
- **𝓡** — regime operator  
- **𝓘** — integrity operator  
- **𝓕** — reinforcement operator  
- **𝓒𝓁** — collapse operator  

Each operator is structural, substrate‑neutral, and regime‑aware.

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# 1. Distinction Operator (𝓓)

### Purpose  
Constructs or refines distinctions within a distinction space.

### Form  
𝓓(distinction_signature) → distinction

### Notes  
- distinctions are structural, not semantic  
- distinctions must be stable under R1  
- no probabilistic interpretation allowed  

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# 2. Signal Operator (𝓢)

### Purpose  
Defines a signal as an operator acting on distinctions.

### Form  
𝓢(operator_signature, distinction_space) → signal_operator

### Notes  
- signals are operators, not messages  
- signals must preserve distinction integrity  
- signals become multi‑layered in R3  

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# 3. Coherence Operator (𝓒)

### Purpose  
Evaluates distinction stability under operator action.

### Form  
𝓒(distinction_space, operator) → coherence_score

### Notes  
- coherence = distinction stability  
- coherence is structural, not probabilistic  
- coherence must be monotonic across R2 → R3  

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# 4. Adjacency Operator (𝓐)

### Purpose  
Measures structural distance between distinctions.

### Form  
𝓐(distinction_A, distinction_B) → adjacency_metric

### Notes  
- adjacency is structural, not probabilistic  
- adjacency must be regime‑stable  
- adjacency supports cross‑layer mapping in R2  

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# 5. Transform Operator (𝓣)

### Purpose  
Applies structural transforms to distinction spaces.

### Form  
𝓣(distinction_space, transform_signature) → transformed_space

### Notes  
- transforms must preserve coherence  
- transforms become dimensional in R3  
- no semantic transforms allowed  

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# 6. Regime Operator (𝓡)

### Purpose  
Transitions distinction behavior across RTT regimes.

### Form  
𝓡(distinction_space, R_i → R_j) → transitioned_space

### Notes  
- transitions must preserve distinction identity  
- transitions must maintain coherence continuity  
- R3 introduces dimensional operators  

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# 7. Integrity Operator (𝓘)

### Purpose  
Checks whether distinctions remain valid after operator action.

### Form  
𝓘(distinction_space) → integrity_report

### Notes  
- checks dimensional consistency  
- checks non‑degeneracy  
- checks operator‑stability  

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# 8. Reinforcement Operator (𝓕)

### Purpose  
Strengthens distinctions through repeated stable operator action.

### Form  
𝓕(distinction_space, operator_history) → reinforced_space

### Notes  
- reinforcement is structural, not semantic  
- reinforcement increases coherence  
- reinforcement must be monotonic  

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# 9. Collapse Operator (𝓒𝓁)

### Purpose  
Classifies distinction failures.

### Form  
𝓒𝓁(distinction_space) → collapse_mode

### Modes  
- **C1:** distinction ambiguity  
- **C2:** dimensional inconsistency  
- **C3:** operator instability  
- **C4:** coherence failure  

### Notes  
Collapse is structural, not probabilistic.

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# Summary

Information Theory operators define:

- distinctions (𝓓)  
- signals as operators (𝓢)  
- coherence (𝓒)  
- adjacency (𝓐)  
- transforms (𝓣)  
- regime transitions (𝓡)  
- integrity (𝓘)  
- reinforcement (𝓕)  
- collapse modes (𝓒𝓁)

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

These operators form the backbone of the Information Theory module.