Trumpet-Coil Phase Accumulation as a Geometric Origin of the Fine Structure Constant in QMU

This work develops a geometric interpretation of the fine-structure constant $\alpha$ as a phase accumulation arising from directed traversal through a structured spatial medium. Rather than treating $\alpha$ as an empirical coupling parameter, it is modeled as a dimensionless ratio that emerges from the interaction between a one-dimensional propagation path and an underlying distributed geometry.

A physically realizable system is introduced in the form of an exponentially tapered helical conductor (the "trumpet coil"), defined by a radius profile
\[
r(z) = r_0 e^{-k z}.
\]
This geometry produces a constant logarithmic gradient,
\[
\frac{d}{dz} \ln r = -k,
\]
which enables uniform phase-slip accumulation along the length of the structure.

A first-order geometric phase law is derived:
\[
d\phi = \alpha \, d\ln \Xi,
\]
where $\Xi$ represents a geometric ratio constructed from the balance of expansive and torsional contributions. Under minimal scaling assumptions, this reduces to
\[
\phi = 2\alpha \ln\left(\frac{r_w}{r_n}\right),
\]
where $r_w$ and $r_n$ are the wide and narrow radii of the tapered structure.

This phase contribution is distinct from conventional transmission-line phase accumulation, which is given by
\[
\phi_{\mathrm{TL}} = \int_0^L \beta(z)\,dz, \quad \beta(z) = \omega \sqrt{L'(z) C'(z)},
\]
and scales primarily with frequency and length. In contrast, the geometric phase term depends only on the logarithmic taper ratio and is predicted to be frequency-invariant to first order.

The total observed phase is therefore decomposed as
\[
\phi_{\mathrm{obs}} = \phi_{\mathrm{TL}} + \phi_{\mathrm{geom}} + \phi_{\mathrm{parasitic}},
\]
where the geometric contribution is isolated experimentally by subtracting the best-fit transmission-line baseline across multiple taper geometries.

A key experimental prediction is a linear relationship:
\[
\frac{d\phi}{d\ln(r_w/r_n)} = 2\alpha,
\]
with a slope on the order of $10^{-2}$ radians. This magnitude is within the resolution of modern vector network analyzers, enabling a practical validation pathway.

A distinguished configuration occurs at
\[
\frac{r_w}{r_n} = e^{\pi/2},
\]
for which the one-pass phase is $\pi \alpha$ and the symmetric round-trip phase is $2\pi \alpha$, suggesting a geometric closure condition.

The results provide a testable framework in which electromagnetic coupling is interpreted as a geometric phase phenomenon. The proposed experiment offers a direct method to determine whether a logarithmic, frequency-independent phase contribution exists beyond conventional distributed electromagnetic behavior.