A Trumpet-Geometry Tesla Resonator to Test the QMU Carrier Mapping

This dataset contains the LaTeX source and supporting materials for a buildable, falsifiable experiment designed to test a core prediction of the Quantum Measurement Units (QMU) framework using a tapered ``trumpet'' Tesla resonator. The apparatus separates a broad, high-current lower region from a narrow, high-potential upper region terminated by a smooth metallic topload, allowing a same-device comparison between an electrostatic top-load channel and a magnetic ground-return channel.

The central revision in this version is the replacement of the earlier direct carrier assignment \(q=C_{\mathrm{top}}V_{\mathrm{pk}}\) and proxy relation \(Q=\kappa I_{\mathrm{pk}}\) with an explicit SI--QMU bridge based on synchronized waveform measurements. The experimentally compared distributed charges are now defined as
\[
q^{(2)}_{e,\mathrm{top}}=\mathrm{ccf}\,C_{\mathrm{top}}V_{\mathrm{pk}},
\qquad
q^{(2)}_{m,\mathrm{gnd}}=\mathrm{ccf}\int_{t_0}^{t_V} I_{\mathrm g}(t)\,dt,
\]
where
\[
\mathrm{ccf}=\frac{{e_{emax}}^{2}}{e}
\]
is the charge conversion factor, \(C_{\mathrm{top}}\) is the in-situ electrostatic capacitance associated with the metallic topload share, \(V_{\mathrm{pk}}\) is the synchronized crest voltage, and the current integral is taken over the dominant charging interval ending at the same crest. The tested QMU mapping is therefore
\[
q^{(2)}_{e,\mathrm{top}}=8\pi\alpha\,q^{(2)}_{m,\mathrm{gnd}},
\]
equivalently
\[
\alpha_{\mathrm{extracted}}
=
\frac{C_{\mathrm{top}}V_{\mathrm{pk}}}
{8\pi\displaystyle\int_{t_0}^{t_V} I_{\mathrm g}(t)\,dt}.
\]

A major structural correction in this version is the explicit distinction between the total terminal capacitance of the assembled resonator,
\[
C_{\mathrm{term}},
\]
and the metallic-topload electrostatic share,
\[
C_{\mathrm{top}}.
\]
This prevents the experiment from collapsing into a trivial restatement of total charge continuity. The conventional electromagnetic benchmark is now written in the factorized form
\[
\alpha_{\mathrm{extracted}}
=
\frac{\eta_{\mathrm{top}}\xi_{\mathrm{wave}}}{8\pi},
\qquad
\eta_{\mathrm{top}}=\frac{C_{\mathrm{top}}}{C_{\mathrm{term}}},
\qquad
\xi_{\mathrm{wave}}=\frac{C_{\mathrm{term}}V_{\mathrm{pk}}}{J_{\mathrm g}},
\]
with
\[
J_{\mathrm g}=\int_{t_0}^{t_V} I_{\mathrm g}(t)\,dt.
\]
Under standard tapered transmission-line modeling, both \(\eta_{\mathrm{top}}\) and \(\xi_{\mathrm{wave}}\) are expected to vary with flare law, topload family, coupling, branch selection, loss, and drive modality. Thus, if \(\alpha_{\mathrm{extracted}}\) remains equal to the fine-structure constant across those controlled changes, the result would exceed the conventional benchmark and support the QMU mapping.

The paper now provides: (i) complete construction specifications for trumpet and cylindrical-control builds, including flare law, turn schedule, topload family, tolerances, and primary geometry; (ii) standard-operating procedures to extract \(C_{\mathrm{term}}\) by low-power VNA fitting and to bracket \(C_{\mathrm{top}}\) through isolated-topload measurement, proximity correction, and differential validation; (iii) synchronized high-power waveform protocols using complex transfer-function de-embedding for the voltage and current channels; (iv) a single-path ground-return requirement so that the measured current integral represents the intended charge transfer; (v) a formal streamer/corona gate to enforce confined, closed-resonator operation; (vi) intentional coupling sweeps and drive-modality replication as explicit nuisance-parameter tests; and (vii) an uncertainty budget propagated directly to \(\alpha_{\mathrm{extracted}}\), with the top-load capacitance bracket treated as the dominant potential systematic.

Implications for the original QMU program remain explicit. If the electrostatic and magnetic distributed charges obey
\[
q^{(2)}_{e,\mathrm{top}}=8\pi\alpha\,q^{(2)}_{m,\mathrm{gnd}},
\]
then the same fixed geometry factor that relates the QMU channel constants,
\[
A_u=16\pi^{2}k_C,
\]
is reflected experimentally in a macroscopic resonant device. In the broader ledger, the channel constants remain linked to the driver through
\[
A_u=\frac{\mathrm{Gforce}\,{\lambda_C}^{2}}{{e_a}^{2}},
\qquad
k_C=\frac{\mathrm{Gforce}\,{\lambda_C}^{2}}{16\pi^{2}e_a^{2}},
\]
so that
\[
A_u=16\pi^{2}k_C,
\qquad
\mathrm{Gforce}=\frac{16\pi^{2}k_C\,{e_a}^{2}}{{\lambda_C}^{2}}.
\]
Accordingly, the trumpet-resonator experiment is positioned not merely as a Tesla-coil geometry study, but as a falsifiable macroscopic probe of the electrostatic--RMFD channel relation within QMU.