# Appendix AG — Derivation 33

Emergent Charge from Phase Winding

Key point. In PBG “charge” is the structural response of a
phase-wound mode in a parity-biased coherence background; it is a
topological current, not a fundamental quantum number.

1 Phase-wound modal ansatz

A stationary envelope with integer winding

implies
$|\mathrm{\nabla }\varphi {|}^2=(n/r{)}^2$.

2 Anchoring cost with a parity-odd cross term

For a non-trivial background coherence bias
${\mathbf{A}}_B\equiv \mathrm{\nabla }\times {\mathbf{B}}_0$
(see Appendix D), the minimal quadratic functional is

• 1st term phase-curvature (sign-blind).
• 2nd term static anchoring (bias energy).
• 3rd term parity-odd cross coupling;
  under $(n\to -n)$ it flips sign because
  $\mathrm{\nabla }\varphi \to -\mathrm{\nabla }\varphi$ while ${\mathbf{A}}_B$ is fixed.

3 Cost difference between opposite windings

Split $\mathcal{C}={\mathcal{C}}_0+n\delta \mathcal{C}$ with

Hence

4 Definition of dimensionless charge

Set

Then $q=n$ exactly — integer quantisation identical to
electric charge units. The physical electron corresponds to $n=-1$,
the positron to $n=+1$.

5 Why the sign matters

• Modes with $n<0$ lower the cross-term → favourable anchoring.
• Modes with $n>0$ raise it → opposite “charge”.
• Global neutrality is automatic because the background bias
  ${\mathbf{A}}_B$ comes from the whole modal sea, enforcing
  $\sum q=0$.

6 Outcomes

• Quantisation: $q/e=n\in \mathbb{Z}$.
• Conservation: winding number cannot change by local
  deformations → charge is conserved.
• Matter / antimatter: opposite $n$ ↔ opposite coupling sign.

No additional gauge field, no imposed symmetry—charge emerges from
phase topology plus a single universal constant $\lambda$.

Appendix AF - Anomalous Magnetic Moment | [Index](./Appendix Master) | Appendix AH - Composite Modes