Charge from Phase Winding

In conventional physics, electric charge is treated as a fundamental quantum number—assigned to particles, conserved in interactions, and coupled to the electromagnetic field.

In modal dynamics, charge is not fundamental. It is a consequence of how a mode’s internal phase structure winds around its coherence boundary.

Phase Topology

A mode is described by:

ψ (x) = ρ (x) e^{i ϕ (x)}

The phase $ϕ (x)$ is not flat—it may wind, twist, or rotate as you move around the mode’s boundary. If you take a closed loop $Γ$ around the mode:

\oint_{Γ} \nabla ϕ \cdot d x = 2 π n

The integer $n$ is the winding number—a topological invariant.

This winding number is not just a mathematical curiosity. It defines the polar asymmetry of the mode’s coherence structure.

Emergent Charge

A mode with nonzero phase winding cannot anchor symmetrically. It produces a directional distortion in the surrounding coherence field $B (x)$ . This asymmetry behaves exactly like electric charge:

It modifies the field around the mode
It interacts with other modes via coherence bias
It repels or attracts based on phase orientation

This is not interaction through a field—it is structural distortion in anchoring cost.

What we observe as “electromagnetic repulsion” is a coherence tension created by overlapping asymmetric phase surfaces.

Sign and Strength

Positive and negative charge correspond to opposite winding orientations
Neutral modes have zero winding (or cancel internally)
The magnitude of the charge corresponds to the stability of the winding—how deeply the mode is anchored with that topology

Charge Conservation

Because phase winding is a topological invariant, it is preserved under smooth evolution. This explains why charge is conserved—no symmetry principle is needed.

Decay and transformation processes preserve winding structure because topology resists fragmentation.

(...as derived in Appendix AG.)

Charge is not a label.
It is a structural asymmetry in phase—an imprint left on the coherence field.