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.