A Josephson–SQUID Barrier-Clock Diagnostic for the Residual Clock–Energy Framework

This record contains Version v0.5r2 of *A Josephson–SQUID Barrier-Clock Diagnostic for the Residual Clock–Energy Framework*. This manuscript is the first device-facing executable diagnostic note following the broader public protocol draft *Prediction and Discriminating Experiments for the Residual Clock–Energy Framework*, Version v0.4r1, DOI: 10.5281/zenodo.19846816.

The purpose of this note is not to propose a new Josephson law, not to claim an empirical detection, and not to replace standard Josephson/SQUID physics. Instead, it builds a residual phase-clock covariance diagnostic on top of standard superconducting phase physics, SQUID flux modulation, RCSJ-style phase dynamics, and tunneling-opacity language.

The central diagnostic map is

$$
u=\frac{\Phi_{\rm ext}}{\Phi_0}
\longmapsto
\left(
I_c(u),
\Gamma_\phi(u),
\frac{\delta\nu_{\rm osc}}{\nu_{\rm osc}}(u),
S_{\phi\phi}(\omega;u)
\right).
$$

Here \(u\) is the normalized external flux, \(I_c\) is the critical current, \(\Gamma_\phi\) is a phase-dephasing rate, \(\delta\nu_{\rm osc}/\nu_{\rm osc}\) is a fractional oscillator or clock-frequency shift, and \(S_{\phi\phi}\) is a phase-noise spectral density.

The Residual Clock–Energy input is represented by the residual phase-tension one-form

$$
\alpha=d\Phi-A,
$$

which is compared with the gauge-invariant superconducting phase difference

$$
\phi
=
\varphi_1-\varphi_2
-
\frac{2\pi}{\Phi_0}
\int_1^2 A\cdot d\ell .
$$

Version v0.5r2 introduces a toy residual flux drive

$$
\mathcal R_J(u)=r_0\sin^2(\pi u),
$$

and studies whether baseline-subtracted observables share the same residual morphology. The diagnostic target is not raw SQUID current modulation, since \(I_c(u)\) already has a standard SQUID interference pattern. The target is multi-observable covariance among baseline-subtracted current correction, dephasing, frequency shift, and phase-noise response.

The manuscript includes:

1. a standard Josephson/SQUID baseline;
2. an RCSJ-style phase-dynamics and noise baseline;
3. a residual clock deformation of Josephson observables;
4. a barrier-clock tunneling layer with WKB recovery and a boundary-layer operational-clock proxy;
5. an executable toy flux grid;
6. morphology scores and an \(H_0/H_1\) covariance diagnostic;
7. nuisance-control and error-budget tables;
8. falsifiability criteria;
9. application outlook for superconducting qubit noise diagnostics, SQUID drift calibration, tunnel-barrier screening, cryogenic RF phase-noise diagnostics, and quantum-metrology error budgets.

The tunneling layer keeps the standard WKB opacity and transmission baseline,

$$
\mathcal R_B=\int_{x_1}^{x_2}\kappa(x)\,dx,
$$

$$
\mathcal T\sim e^{-2\mathcal R_B}.
$$

The first-pass traversal-time observable is the phase-time or Wigner–Smith proxy,

$$
\tau_\varphi(E)=\hbar\frac{\partial}{\partial E}\arg t(E).
$$

For real Josephson junctions, \(\mathcal R_B\) is treated as an effective opacity ledger rather than as a literal single-particle WKB parameter unless a specific junction model justifies that reduction.

The toy curves in this record should be read as positive-control injections. Their purpose is to define the residual morphology that a real experiment would attempt to recover after nuisance projection. They are not empirical evidence for the Residual Clock–Energy framework.

The included reproducibility materials provide the LaTeX source, PDF, toy flux-grid CSV file, generated figures, and the Python figure-generation script used for the diagnostic plots.

The next intended step is a v0.6 sensitivity version with synthetic noise injection, channel-dependent detection thresholds, and explicit calibration targets such as

$$
\Delta\Gamma_{\phi,\min},
\quad
\Delta(\delta\nu/\nu)_{\min},
\quad
S_{\phi\phi}^{\min},
\quad
M_{\min}.
$$