Calibration-Ready Josephson–SQUID Diagnostics for Residual Barrier Perturbations: Fail-Closed Nuisance Conditioning, Multi-Witness Noise Rejection, and Ramsey Closure

This work develops a device-parameterized, calibration-ready synthetic diagnostic for weak residual barrier perturbations in a split-junction Josephson circuit. It is the experimental-methodology companion to the Residual Clock–Energy program, but it does not report an experimental detection and does not derive a new Josephson relation. The device-facing hypothesis is $$E_J(u,t) = E_J^{(0)}(u) \\, [1 + \\zeta(u,t)],$$ where $\\zeta$ is a weak latent barrier perturbation. Feynman–Hellmann differentiation fixes the first-order frequency response, $$\\delta \\ln \\omega_{01}(u) = \\kappa_J(u) \\, \\zeta(u) + O(\\zeta^2),$$ replacing the free frequency-response coefficient used in the earlier morphology-only diagnostic. The analysis uses a frozen asymmetric split-transmon model, parity-protected paired-flux observables, a finite-calibration nuisance manifold, generalized matched filtering, bounded systematic leakage, and a predeclared false-positive reserve. Random calibration uncertainty is marginalized into an effective covariance, while bounded systematic uncertainty is propagated into a worst-case residual-amplitude bias certificate. The final synthetic ledger adopts a 1% false-positive target and a 5% reserve policy. With the final even-pickup bound $$\\vert{}p_{\\mathrm{even}}\\vert{} \\le 1.15 \\times 10^{-7},$$ it gives $$\\mathrm{VIF}_{\\mathrm{eff}} = 1.087, \\qquad B_* = 8.743 \\times 10^{-7},$$ a nominal false-positive reserve of approximately $6.07\\%$, and a robust $95\\%$ recovery floor $$a_{\\min,95}^{\\mathrm{robust}} \\simeq 7.29 \\times 10^{-6}.$$ At the frozen half-flux operating point this corresponds to a synthetic frequency-response floor of about $15.5\\text{ kHz}$. These values are calibration-derived sensitivity requirements of the declared synthetic model, not predictions of residual physics. The dynamic nuisance branch replaces a generic quasiparticle-like Ornstein–Uhlenbeck process with trapping–recombination–generation dynamics for the normalized quasiparticle density $x_{\\mathrm{qp}}$. The same latent density drives both the qubit-frequency nuisance and a $T_1$-derived witness. In the declared synthetic population, multi-witness conditioning removes about 99% of the modeled quasiparticle-induced frequency-noise variance. Ramsey closure is treated conservatively. Gaussian PSD-to-Ramsey reconstruction is accepted only over the predeclared short coherent window $$t \\le 50 \\, \\mu\\mathrm{s},$$ while longer-time quasiparticle coherence is left to direct time-domain or non-Gaussian treatment. A separate exact quasi-static calculation provides a guardrail for quadratic sweet-spot flux noise. The diagnostic is explicitly fail-closed. Static spectroscopy alone cannot determine the microscopic origin of a perturbation that acts through the same effective Josephson-energy channel, and an unmodeled nuisance exactly parallel to the candidate template remains non-identifiable. Candidate admission therefore requires frozen circuit transfer, parity checks, calibrated thermal/quasiparticle/pickup controls, systematic-bias and reserve gates, independent witness conditioning, and confidence-calibrated recovery. This paper is the third part of a public three-paper sequence: Paper I: Residual Clock Dynamics: Operational-Time Rescaling and Effective Relaxation in a Minimal $U(1)$ Model, Version v1.0, Zenodo DOI: 10.5281/zenodo.22735204. Paper II: Field-Theoretic Residual Clock Response and the Criterion for a Common Operational Time, Version v1.0, Zenodo DOI: 10.5281/zenodo.22739058. Paper II derives the conditional parent-model chain $$X \\longrightarrow \\rho_X \\longrightarrow \\omega_C(X),$$ whereas the present work assumes the separate device-level chain $$\\zeta \\longrightarrow E_J \\longrightarrow \\omega_{01}.$$ No microscopic derivation of $X \\longrightarrow \\zeta$ is claimed here. Establishing or ruling out that theory-to-device matching remains an open problem. The accompanying reproducibility package contains the manuscript source, scripts, numerical tables, figures, environment specification, execution instructions, and checksums needed to reproduce the synthetic calculations.