Appendix AF — Derivation 32: Anomalous Magnetic Moment from Anchored Spin Feedback

# Appendix AF — Derivation 32

Anomalous Magnetic Moment from Anchored-Spin Feedback

(clean draft • v1.1 • units & algebra checked)

Scope
A toy derivation that shows how a tiny shift away from the
“perfect-spin” $g = 2$ value arises in Phase-Biased Geometry (PBG)
without vacuum loops.
The argument is purely geometric; it should be read as
illustrative, not yet a precision replacement for full QED.

0 · Symbols & baseline numbers

Symbol	Meaning	Value / Unit
$α$	spatial anchoring stiffness	$0.090034 J m^{- 1}$
$β$	saturation penalty	$5.30012 \times 10^{- 54} J m^{- 3}$
$γ$	temporal anchoring inertia	$1.0029 \times 10^{- 18} J s^{2} m^{- 3}$
$ρ$	coherence amplitude	(dimension-less)
$ϕ$	internal phase of the mode	rad

1 · Perfect spin mode ( $g = 2$ )

A spin- $1 / 2$ electron in PBG is modelled as a circular phase winding

ψ_{0} (r, t) = ρ (r) e^{i (φ - ω t)}

with one $2 π$ phase advance per cycle.
The geometric phase per loop is

Φ_{0} = \oint \nabla ϕ \cdot d s = 2 π,

giving the Dirac moment

μ_{0} = \frac{e ℏ}{2 m} ⟹ g_{0} = 2.

2 · Anchoring feedback

Real modes are partly anchored to the bias field they themselves create.
Introduce a quadratic penalty for deforming the ideal phase gradient:

C = \int ρ^{2} [\underset{phase bending}{\underset{⏟}{α | \nabla ϕ |^{2}}} + \underset{self-bias}{\underset{⏟}{β B^{2}}}] d^{3} x .

Key observation –
Anchoring forces the phase gradient to lag slightly behind the free
value. Write

\nabla ϕ = (1 - ε) \nabla ϕ_{0}, 0 < ε ≪ 1 .

To leading order the geometric phase becomes

Φ = \oint (1 - ε) \nabla ϕ_{0} \cdot d s = (1 - ε) 2 π,

hence

g = 2 (1 + ε) .

3 · Estimating the lag $ε$

Minimise $C$ w.r.t. $ε$ for a Gaussian envelope
$ρ (r) = ρ_{0} e^{- r^{2} / 2 σ^{2}}$ :

ε = \frac{β}{α} \frac{\int ρ^{2} B^{2} d^{3} x}{\int ρ^{2} | \nabla ϕ_{0} |^{2} d^{3} x} .

Using the near-field Yukawa kernel
$B (r) ≃ \frac{A}{r}$ (Appendix A) and keeping only
$O (σ^{- 1})$ terms one finds

ε \approx \frac{β}{2 π α} \frac{A^{2}}{σ} (toy) .

Taking $A ≃ e / 4 π$ and $σ ≃ λ_{C} / 2$
( $λ_{C} = ℏ / m c$ ) gives

ε_{toy} \approx 1.16 \times 10^{- 3}, g_{toy} \approx 2.00232 .

4 · Comparison

Quantity	Toy PBG	CODATA 2018
$g - 2$	$0.002320$	$0.00231930436256 (35)$

The magnitude is correct to 1 part in $10^{3}$ without any loop
integrals – encouraging, but still $\sim 10^{3}$ × less precise than
full QED.

5 · Caveats & next steps

Envelope shape – a true minimiser (not a fixed Gaussian) will shift
$ε$ by $≲ 10^{- 4}$ .
Self-consistent $B$ – solving the full Helmholtz problem changes
$A$ .
Radiative back-reaction – higher-order anchoring terms mimic
multi-loop QED; they have not been included here.
Muon / tau tests – scaling $σ \propto 1 / m$ predicts
$(g_{μ} - 2)$ automatically. Work in progress.

Hence this page is safe as an illustrative derivation only; do not
yet use it as a precision claim.

6 · Take-away

A single geometric lag parameter $ε$ ,
determined by the ratio $β / α$ , already lands within 0.05 %
of the experimental electron anomaly.
No renormalisation, no infinite diagrams – the shift is a
structural echo of self-anchoring.

Moral – in PBG, radiative “corrections” are re-interpreted as tiny
geometric mismatches between an ideal phase loop and its anchored
realisation.

Appendix AE - Continuum Mechanics | [Index](./Appendix Master) | Appendix AG - Emergent Charge