Appendix AN — Expanded Electromagnetism and Gauge Symmetry

# Appendix AN — Electromagnetism and Gauge Structure in PBG

AN.1 — Motivation: Beyond Fields and Photons

Conventional physics describes electromagnetism through classical fields and quantum photons, treating “charge” and “force” as primitive properties.
Phase-Biased Geometry (PBG) eliminates these ontologies.
All electromagnetic phenomena emerge from the geometry of phase in a coherence medium, with interaction, propagation, and quantisation as necessary consequences of modal structure and anchoring cost.

A coherence mode in PBG is defined by

ψ (\vec{x}, t) = ρ (\vec{x}, t) e^{i φ (\vec{x}, t)}

Dynamical evolution follows from minimising the anchoring cost:

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

where:

$α = 0.090034$ J $/ m$ (anchoring stiffness)
$β = 5.30012 \times 10^{- 54}$ J/m $^{3}$ (envelope penalty)

Charge arises as a phase winding—an anchoring asymmetry, not a fundamental property.

AN.3 — Gauge Invariance as Structural Freedom

Electromagnetic gauge invariance (U(1)) in PBG:

Transformation: $ψ (\vec{x}) \to ψ^{'} (\vec{x}) = ψ (\vec{x}) e^{i α (\vec{x})}$
The anchoring cost is unchanged for slow variations of $α (\vec{x})$ , reflecting the structural freedom of local phase without energy cost.

AN.4 — Observable Electromagnetic Structure in PBG

Charge and Coulomb Law

Positive and negative charges correspond to opposite senses of phase winding ( $q$ ), e.g., $φ = q θ$ .
The Coulomb force emerges as bias-following motion in the gradient of the anchoring cost, yielding an effective $1 / r$ law between spherically symmetric modes.

Photon Structure

A photon is a propagating modal phase packet, not a particle.
The propagation speed is set by the ratio of anchoring constants: $c^{2} = \frac{2 α}{γ}$ where $γ = 1.000228 \times 10^{- 18}$ J·s $^{2}$ /m $^{3}$ .

Maxwell’s Equations

Maxwell’s equations are statistical summaries that arise in the limit of dense modal ensembles, not as fundamental laws.

AN.5 — Structural Analogues: Mapping EM to PBG

Classical / QED Concept	PBG Structural Replacement
Electric field ( $\vec{E}$ )	Phase gradient across coherence field
Magnetic field ( $\vec{B}$ )	Spatial twist in phase topology
Charge ( $e$ )	Anchoring asymmetry / phase winding number
Vector potential ( $\vec{A}$ )	Local modal phase offset (structural, not dynamical)
Gauge symmetry (U(1))	Anchoring cost invariance under local phase redefinition
Photon	Modal phase packet (coherence event)
Coulomb's Law	Anchoring cost gradient between phase-wound modes
Maxwell’s equations	Statistical average of cost-minimising modal phase flows
Fine structure constant ( $α_{f s}$ )	Ratio of coherence costs for phase-winding and modal interaction

AN.6 — Notation Caution: Two Alphas

Caution:
The universal anchoring constant ( $α$ , with physical units) is not the same as the dimensionless fine structure constant ( $α_{f s}$ ).

$α$ : Sets phase stiffness in the modal cost functional.

$α_{f s}$ : Sets the observed electromagnetic coupling strength; emerges as a ratio of coherence costs in PBG.

This distinction is maintained throughout all formulae.

AN.7 — Fine Structure Constant in PBG

The fine structure constant,

α_{f s} = \frac{e^{2}}{4 π ϵ_{0} ℏ c} \approx \frac{1}{137.036}

is dimensionless and quantifies electromagnetic coupling strength.

In PBG, $α_{f s}$ emerges as:

The ratio of the coherence cost for a phase-wound mode (charge) to the cost of free phase propagation,
Or, equivalently, as the ratio of anchoring cost for an azimuthal (charged) winding to the baseline modal energy.

A formal derivation shows:

α_{f s} \sim \frac{anchoring cost for phase winding}{ℏ c / a}

where $a$ is the characteristic mode radius (e.g., the electron’s Compton wavelength).

AN.8 — Calibrated Universal Constants

Constant	Value	Units
$α$	$0.090034$	J/m
$β$	$5.30012 \times 10^{- 54}$	J/m $^{3}$
$γ$	$1.000228 \times 10^{- 18}$	J·s $^{2}$ /m $^{3}$

AN.9 — Why PBG Succeeds

Electromagnetism in PBG is coherence mechanics.
Every classical and quantum electromagnetic observable is structurally replaced by phase, cost, and winding—anchored by universal constants and free from classical metaphors or metaphysical fields.