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

The document includes ready-to-build dimensions, BOM guidance, and calibration steps for the total terminal capacitance \(C_{\mathrm{term}}\), the bracketed in-situ metallic-topload capacitance \(C_{\mathrm{top}}\), synchronized voltage and ground-return current waveforms, and coupling control.

Data-quality controls require operation with no visible streamers; runs with corona, broadband RF bursts, ambiguous crest timing, branch hopping, or invalid current-integration windows are gated out.

The uncertainty propagation reflects the linear extracted relation
\[
\alpha_{\mathrm{extracted}}
=
\frac{C_{\mathrm{top}}V_{\mathrm{pk}}}{8\pi J_{\mathrm g}},
\qquad
J_{\mathrm g}=\int_{t_0}^{t_V} I_{\mathrm g}(t)\,dt,
\]
so the dominant relative-error contributions arise from \(C_{\mathrm{top}}\), \(V_{\mathrm{pk}}\), \(J_{\mathrm g}\), de-embedding, coupling, drive modality, and repeatability. There is no overall factor of \(4\), since the revised experiment no longer uses the earlier square-law proxy relation.

The benchmark derived from Maxwell’s equations and tapered transmission-line modeling provides the null-hypothesis geometry dependence in the form
\[
\alpha_{\mathrm{extracted}}=\frac{\eta_{\mathrm{top}}\xi_{\mathrm{wave}}}{8\pi},
\]
against which the geometry-, coupling-, and drive-independent QMU mapping is tested.

Acceptance criteria require flatness of \(\alpha_{\mathrm{extracted}}\) versus drive power, robustness under intentional coupling variation and drive-modality changes, and invariance across trumpet and control geometries within the stated uncertainty budget.

This revision strengthens the experimental definition, physical orientation, and falsification logic of the trumpet-geometry Tesla resonator.

The apparatus orientation is now stated explicitly: the broad lower end is the primary-coupled, high-current magnetic region, while the narrow upper end is the high-potential electrostatic region. This removes ambiguity about the intended current and potential distribution along the trumpet secondary.

The conventional electromagnetic benchmark has also been reformulated to avoid a tautological comparison. The measured quantities

$$
\eta_{\mathrm{top}}=\frac{C_{\mathrm{top}}}{C_{\mathrm{term}}}
$$

and

$$
\xi_{\mathrm{wave}}
=
\frac{C_{\mathrm{term}}V_{\mathrm{pk}}}{J_{\mathrm{g}}}
$$

satisfy the algebraic identity

$$
\alpha_{\mathrm{extracted}}
=
\frac{\eta_{\mathrm{top}}\xi_{\mathrm{wave}}}{8\pi}.
$$

This identity is now treated only as a decomposition of the measured estimator, not as an independent conventional prediction.

The conventional benchmark is instead defined through a Maxwell-based tapered transmission-line model that independently predicts

$$
\xi_{\mathrm{EM}}
=
\frac{C_{\mathrm{term}}V_{\mathrm{pk,EM}}}
{J_{\mathrm{g,EM}}},
$$

and therefore

$$
\alpha_{\mathrm{EM}}
=
\frac{\eta_{\mathrm{top}}\xi_{\mathrm{EM}}}{8\pi}.
$$

The primary experimental comparison is now between the directly measured quantity

$$
\alpha_{\mathrm{extracted}}
=
\frac{C_{\mathrm{top}}V_{\mathrm{pk}}}
{8\pi J_{\mathrm{g}}}
$$

and the independently calculated conventional prediction $\alpha_{\mathrm{EM}}$.

For fixed geometry and topload, the QMU mapping

$$
\alpha_{\mathrm{extracted}}=\alpha
$$

also requires

$$
\xi_{\mathrm{wave}}
=
\frac{8\pi\alpha}{\eta_{\mathrm{top}}}.
$$

This makes the coupling and drive-modality tests more restrictive: if the electrostatic partition remains fixed, both $\alpha_{\mathrm{extracted}}$ and the measured waveform factor $\xi_{\mathrm{wave}}$ must remain invariant within uncertainty.

The uncertainty treatment has been corrected accordingly. Because $C_{\mathrm{term}}$ appears inversely in $\eta_{\mathrm{top}}$ and directly in $\xi_{\mathrm{wave}}$, those two measured quantities are correlated and must not be propagated as independent uncertainties. The primary uncertainty budget therefore remains based on the direct estimator

$$
\alpha_{\mathrm{extracted}}
=
\frac{C_{\mathrm{top}}V_{\mathrm{pk}}}
{8\pi J_{\mathrm{g}}},
$$

while covariance is included when the product form is used diagnostically.

The experimental protocol has also been refined to preserve the operating boundary conditions during low-power VNA characterization, to require the primary at the broad end of the trumpet, and to distinguish intended primary and ground structures from unintended nearby metal.

The estimated secondary turn count has been corrected from approximately 257 turns to approximately 272 turns by integrating the specified piecewise winding-pitch schedule:

$$
N
=
\int_0^H \frac{dz}{p(z)}
\approx 272.
$$

The power-invariance criterion has also been replaced with a statistically defined slope test rather than the previous absolute-derivative condition.

Finally, the interpretation section now distinguishes three possible outcomes: QMU support, conventional-benchmark support, and an indeterminate result when the two predictions overlap within combined experimental and model uncertainty.

These revisions preserve the proposed QMU charge-partition mapping

$$
q^{(2)}_{e,\mathrm{top}}
=
8\pi\alpha\,q^{(2)}_{m,\mathrm{gnd}}
$$

while making the competing conventional prediction genuinely independent, the metrology internally consistent, and the experimental falsification criteria more rigorous.