Appendix AG — Derivation 33: Emergent Charge from Modal Phase Winding

In Phase-Biased Geometry (PBG), charge is not a fundamental quantity or quantum number. It arises as a structural asymmetry in the coherence and anchoring behaviour of modes with non-zero phase winding.

This derivation demonstrates how the azimuthal phase topology of a mode leads to a persistent anchoring cost imbalance—experienced observationally as “charge”.

Consider a coherent modal structure with azimuthal phase winding:

ϕ (θ) = n θ

where:

( \theta \in [0, 2\pi) )
( n \in \mathbb{Z} ) is the winding number
( \phi ) is the internal phase of the mode

Such a structure cannot unwind without decohering, and its coherence envelope must accommodate the persistent angular gradient.

2. Anchoring Cost Functional

The general cost of maintaining a coherent modal field is:

C [ψ] = \int ρ^{2} (α | \nabla ϕ |^{2} + β B^{2} [ψ]) d^{3} x

For a mode with circular winding, the dominant cost comes from the azimuthal gradient:

| \nabla ϕ |^{2} = {(\frac{n}{r})}^{2}

This results in a centrally peaked anchoring cost that resists coherent compression. The cost is asymmetric under reflection of phase gradient direction (clockwise vs counter-clockwise), but since ( |\nabla \phi|^2 ) is symmetric in ( n ), an additional asymmetry is needed.

3. Topological Anchoring Asymmetry

If the coherence field ( B[\psi] ) responds differently to positive vs negative windings—due to background coherence structure or phase alignment constraints—then:

C_{+} \neq C_{-}

This asymmetry is structural: it arises from how phase-aligned modal currents interact with their environment. For example, in a vacuum biased by previous coherence gradients, modes with phase winding in one direction may find easier anchoring paths than the other.

4. Definition of Charge

We define electric charge in PBG as the emergent structural asymmetry resulting from persistent phase winding in the presence of anchoring bias.

Let:

q = \frac{δ C}{δ ϕ}

That is, charge is the coherence cost derivative with respect to phase winding structure.

This yields a discrete, polarised value tied to:

The sign of the winding number ( n )
The coherence alignment of the surrounding field

5. Summary

Charge is not fundamental, but emergent
It arises from anchoring asymmetry in topologically non-trivial phase structures
This naturally explains:
- Charge conservation (topological constraint)
- Charge quantisation (winding number)
- Matter/antimatter symmetry (opposite windings)

Implications

This derivation eliminates the need for a conserved scalar “charge” field or source term. Instead, the entire electromagnetic structure in PBG arises from modal coherence, winding topology, and anchoring asymmetry.

Appendix AF | [Index](./Appendix Master) | Appendix AH