Examples — Electromagnetism
TriadicFrameworks /docs/theories/electromagnetism/examples.md#
These examples illustrate Electromagnetism as 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.
All examples avoid force metaphors, particle‑centric drift, and teleology.
1. Electric Divergence Example (𝓓ᴱ)#
Goal#
Relate electric field divergence to charge density.
Input#
E = electric_field
ρ = charge_density
Operation#
divE = 𝓓ᴱ(E) = ∇·E
Interpretation#
- divergence expresses field‑source structure
- ρ/ε₀ is a source operator, not a particle property
- no action‑at‑a‑distance framing
2. Magnetic Divergence Example (𝓓ᴮ)#
Goal#
Enforce magnetic coherence.
Input#
B = magnetic_field
Operation#
divB = 𝓓ᴮ(B) = ∇·B
Interpretation#
- ∇·B = 0 is a coherence constraint
- expresses structural consistency of B
- no monopole metaphors
3. Electric Curl Example (𝓒ᴱ)#
Goal#
Relate electric field rotation to changing magnetic fields.
Input#
E = electric_field
B = magnetic_field
Operation#
curlE = 𝓒ᴱ(E) = ∇×E = −∂B/∂t
Interpretation#
- curl is a structural operator, not a force
- time‑variation is geometric, not teleological
4. Magnetic Curl Example (𝓒ᴮ)#
Goal#
Relate magnetic field rotation to current and changing electric fields.
Input#
B = magnetic_field
J = current_density
Operation#
curlB = 𝓒ᴮ(B) = ∇×B = μ₀J + μ₀ε₀∂E/∂t
Interpretation#
- current is a source operator, not a particle stream
- curl expresses field rotation, not mechanical force
5. Charge‑Source Example (𝓢ᶜʰ)#
Goal#
Define charge as a divergence source.
Input#
ρ = charge_density
Operation#
E_source = 𝓢ᶜʰ(ρ)
Interpretation#
- charge modifies divergence structure
- no particle‑centric framing
6. Current‑Source Example (𝓢ᶜᵘʳ)#
Goal#
Define current as a curl source.
Input#
J = current_density
Operation#
B_source = 𝓢ᶜᵘʳ(J)
Interpretation#
- current modifies curl structure
- structural, not mechanical
7. Wave Propagation Example (𝓦)#
Goal#
Propagate EM fields through space‑time.
Input#
E = electric_field
B = magnetic_field
geometry = flat_space
Operation#
E', B' = 𝓦(E, B)
Interpretation#
- light = self‑coherent field propagation
- no medium (ether) metaphors
- propagation respects geometry
8. Field‑Tensor Example (𝓕)#
Goal#
Unify E and B into a geometric object.
Input#
E = electric_field
B = magnetic_field
Operation#
F_uv = 𝓕(E, B)
Interpretation#
- required for R3 (geometry‑coupled EM)
- supports GR and QFT integration
- coherence evaluated via invariants
9. Coherence Evaluation Example (𝓒ₒₕ)#
Goal#
Evaluate electromagnetic coherence.
Input#
E = electric_field
B = magnetic_field
geometry = flat_space
Operation#
coh = 𝓒ₒₕ(E, B, geometry)
Interpretation#
Coherence requires:
- divergence consistency
- curl consistency
- propagation stability
- geometric compatibility
10. Regime Transition Example (𝓡𝓮𝓰)#
Goal#
Transition EM behavior from R1 → R2.
Input#
field_state = static_configuration
Operation#
state_R2 = 𝓡𝓮𝓰(field_state, R1 → R2)
Interpretation#
- time‑variation activates
- dynamic curl operators engage
- wave propagation emerges
11. Collapse Classification Example (𝓒𝓁)#
Goal#
Classify electromagnetic failure.
Input#
field_state = unstable_field
Operation#
mode = 𝓒𝓁(field_state)
Possible Outputs#
- EM1: divergence collapse
- EM2: curl collapse
- EM3: propagation collapse
- EM4: source collapse
- EM5: geometry collapse
Interpretation#
Collapse is structural, not force‑based.
Summary#
These examples show Electromagnetism as:
- field‑first
- operator‑driven
- coherence‑based
- regime‑aware
- geometry‑compatible
- zero drift
Electromagnetism = coherent field behavior.
Maxwell operators = structural constraints.
Light = self‑consistent field propagation.