# Operators — Electromagnetism  
### TriadicFrameworks /docs/theories/electromagnetism/operators.md

Electromagnetism (EM) in TriadicFrameworks is a **field‑coherence
theory**, not a force‑centric mechanism and not a particle‑first
narrative.

EM = **coherent behavior of the electromagnetic field**.  
Maxwell operators = **structural constraints**, not force laws.  
Light = **self‑consistent field propagation**.

This file defines the canonical operators for Electromagnetism across
R1 → R3.

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# Operator List

The core operators are:

- **𝓓ᴱ** — electric divergence operator  
- **𝓓ᴮ** — magnetic divergence operator  
- **𝓒ᴱ** — electric curl operator  
- **𝓒ᴮ** — magnetic curl operator  
- **𝓢ᶜʰ** — charge‑source operator  
- **𝓢ᶜᵘʳ** — current‑source operator  
- **𝓦** — wave propagation operator  
- **𝓕** — field‑tensor operator  
- **𝓒ₒₕ** — coherence operator  
- **𝓡𝓮𝓰** — regime transition operator  
- **𝓒𝓁** — collapse operator  

Each operator is structural, non‑teleological, and field‑first.

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# 1. Electric Divergence Operator (𝓓ᴱ)

### Purpose  
Relate electric field divergence to charge density.

### Form  
𝓓ᴱ(E) = ∇·E = ρ/ε₀

### Notes  
- charge is a **source operator**, not a particle  
- divergence is a **coherence constraint**  
- no force‑centric framing  

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# 2. Magnetic Divergence Operator (𝓓ᴮ)

### Purpose  
Enforce magnetic coherence.

### Form  
𝓓ᴮ(B) = ∇·B = 0

### Notes  
- expresses magnetic field coherence  
- no magnetic monopole metaphors  
- structural constraint, not a physical “rule”  

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# 3. Electric Curl Operator (𝓒ᴱ)

### Purpose  
Relate electric field rotation to changing magnetic fields.

### Form  
𝓒ᴱ(E) = ∇×E = −∂B/∂t

### Notes  
- curl is a **structural operator**  
- no “induced force” metaphors  
- time‑variation is geometric, not teleological  

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# 4. Magnetic Curl Operator (𝓒ᴮ)

### Purpose  
Relate magnetic field rotation to current and changing electric fields.

### Form  
𝓒ᴮ(B) = ∇×B = μ₀J + μ₀ε₀∂E/∂t

### Notes  
- current is a **source operator**, not a particle stream  
- curl expresses field rotation, not force  

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# 5. Charge‑Source Operator (𝓢ᶜʰ)

### Purpose  
Define charge as a divergence source.

### Form  
𝓢ᶜʰ(ρ) → divergence contribution to E

### Notes  
- charge is a **field‑source**, not a particle property  
- no action‑at‑a‑distance framing  

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# 6. Current‑Source Operator (𝓢ᶜᵘʳ)

### Purpose  
Define current as a curl source.

### Form  
𝓢ᶜᵘʳ(J) → curl contribution to B

### Notes  
- current is a **field‑source**, not a flow of particles  
- structural, not mechanical  

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# 7. Wave Propagation Operator (𝓦)

### Purpose  
Propagate EM fields through space‑time.

### Form  
𝓦(E, B) = wave(E, B)  
Derived from Maxwell operators.

### Notes  
- light = self‑coherent field propagation  
- no medium (ether) metaphors  
- propagation must respect geometry  

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# 8. Field‑Tensor Operator (𝓕)

### Purpose  
Unify E and B into a geometric object.

### Form  
𝓕(Fᵤᵥ) = EM field tensor

### Notes  
- required for R3 (geometry‑coupled EM)  
- supports GR and QFT integration  
- coherence evaluated via invariants  

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# 9. Coherence Operator (𝓒ₒₕ)

### Purpose  
Evaluate electromagnetic coherence.

### Form  
𝓒ₒₕ(E, B, geometry) → coherence_score

### Notes  
Coherence requires:

- divergence consistency  
- curl consistency  
- propagation stability  
- geometric compatibility  

No force‑centric metrics.

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# 10. Regime Transition Operator (𝓡𝓮𝓰)

### Purpose  
Transition EM behavior across R1 → R3.

### Form  
𝓡𝓮𝓰(field_state, Rᵢ → Rⱼ) → transitioned_state

### Notes  
- R1: classical field stability  
- R2: dynamic propagation  
- R3: geometry‑coupled, multi‑scale EM  
- transitions must preserve coherence  

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# 11. Collapse Operator (𝓒𝓁)

### Purpose  
Classify electromagnetic failure modes.

### Form  
𝓒𝓁(field_state) → collapse_mode

### Modes  
- **EM1:** divergence collapse  
- **EM2:** curl collapse  
- **EM3:** propagation collapse  
- **EM4:** source collapse  
- **EM5:** geometry collapse  

Collapse is structural, not force‑based.

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# Summary

Electromagnetic operators define:

- divergence structure (𝓓ᴱ, 𝓓ᴮ)  
- curl structure (𝓒ᴱ, 𝓒ᴮ)  
- source structure (𝓢ᶜʰ, 𝓢ᶜᵘʳ)  
- propagation (𝓦)  
- geometric unification (𝓕)  
- coherence evaluation (𝓒ₒₕ)  
- regime transitions (𝓡𝓮𝓰)  
- collapse modes (𝓒𝓁)

Electromagnetism = **coherent field behavior**, not force.  
Maxwell operators = **structural constraints**, not particle rules.  
Light = **self‑consistent field propagation**.