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

Byoungwoo Lee

Published Apr 28, 2026• 0 citations
Publisher

Abstract

This record contains Version v0.6r2 of *A Josephson–SQUID Barrier-Clock Diagnostic for the Residual Clock–Energy Framework*. This version is a synthetic-noise sensitivity and application-readiness update to the earlier Josephson–SQUID executable diagnostic line. This manuscript is not an empirical detection paper, not a proposal for a new Josephson law, and not a replacement for standard Josephson/SQUID physics. Instead, it develops a residual phase-clock covariance diagnostic built on standard Josephson/SQUID baseline physics, RCSJ-style phase dynamics, tunneling-opacity language, and mock-data sensitivity calibration. The diagnostic is organized around the flux-control parameter $$u=\frac{\Phi_{\rm ext}}{\Phi_0},$$ and the observable vector $$u\longmapsto\left(I_c(u),\Gamma_\phi(u),\frac{\delta\nu_{\rm osc}}{\nu_{\rm osc}}(u),S_{\phi\phi}(\omega;u)\right).$$ Here \(I_c\) 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 $$\phi=\varphi_1-\varphi_2-\frac{2\pi}{\Phi_0}\int_1^2 A\cdot d\ell .$$ The earlier v0.5r2 version introduced a positive-control toy residual drive, $$\mathcal R_J(u)=r_0\sin^2(\pi u),$$ and used it as a morphology target for baseline-subtracted Josephson/SQUID observables. Version v0.6r2 extends this line by injecting synthetic noise and nuisance morphologies into mock observations, $$y_{k,i}^{\rm obs}(A)=A\beta_k g_i+n_{k,i}+h_{k,i},$$ where \(g_i\) is the centered residual morphology, \(n_{k,i}\) is a synthetic noise term, and \(h_{k,i}\) is a nuisance contribution. The central sensitivity outputs are the detection probability, $$P_{\rm det}(A)=\frac{1}{N_{\rm MC}}\sum_{m=1}^{N_{\rm MC}}\mathbf 1[\text{trial }m\text{ passes}],$$ the minimum detectable amplitude, $$A_{\min}=\inf\{A:P_{\rm det}(A)\ge0.95\},$$ and the false-positive rate, $$P_{\rm FP}=P_{\rm det}(0).$$ Version v0.6r2 includes: 1. synthetic observation models for residual and nuisance channels;2. white-noise, high-noise, correlated-drift, heating-mimic, and mixed-nuisance scenarios;3. morphology-score and signal-to-noise recovery criteria;4. a multi-channel admission rule requiring at least two non-current channels to pass;5. matched-control subtraction for heating-like nuisance;6. false-positive and detection-probability curves;7. sensitivity tables and CSV outputs;8. an application-readiness ladder from toy positive control to device-screening protocol;9. laboratory deployment requirements for a future calibrated experiment;10. device-facing output interpretation tables. A key outcome of the synthetic study is that heating-like nuisance can overlap strongly with the residual morphology and can therefore create false-positive hazards if not controlled. The v0.6r2 update explicitly treats this issue by introducing matched-control subtraction and by distinguishing raw nuisance-contaminated recovery from controlled recovery. The application-readiness ladder used in this version is: - L0: toy positive-control morphology;- L1: synthetic-noise recovery and false-positive stress test;- L2: calibrated synthetic noise using measured device noise floors;- L3: laboratory pilot with real Josephson/SQUID data;- L4: device-screening protocol for phase-noise and barrier-quality diagnostics. The current v0.6r2 manuscript is positioned at L1. It is therefore a synthetic sensitivity and application-readiness note, not a calibrated laboratory demonstration. The included files provide the PDF manuscript, LaTeX source, figure-generation script, synthetic sensitivity summary CSV, detection-curve CSV, matched-control example CSV, simulation metadata JSON, and a source bundle for reproducibility. This work is a companion device-facing diagnostic note following the broader protocol record *Prediction and Discriminating Experiments for the Residual Clock–Energy Framework*, Version v0.4r1, DOI: 10.5281/zenodo.19846816.

DOI
10.5281/zenodo.19846815

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This record contains Version v0.6r2 of A Josephson–SQUID Barrier-Clock Diagnostic for the Residual Clock–Energy Framework. This version is a synthetic-noise sensitivity and application-readiness update to the earlier Josephson–SQUID executable diagnostic line. This manuscript is not an empirical detection paper, not a proposal for a new Josephson law, and not a replacement for standard Josephson/SQUID physics. Instead, it develops a residual phase-clock covariance diagnostic built on standard Josephson/SQUID baseline physics, RCSJ-style phase dynamics, tunneling-opacity language, and mock-data sensitivity calibration. The diagnostic is organized around the flux-control parameter u=ΦextΦ0,u=\frac{\Phi_{\rm ext}}{\Phi_0},u=Φ0​Φext​​, and the observable vector u⟼(Ic(u),Γϕ(u),δνoscνosc(u),Sϕϕ(ω;u)).u\longmapsto\left(I_c(u),\Gamma_\phi(u),\frac{\delta\nu_{\rm osc}}{\nu_{\rm osc}}(u),S_{\phi\phi}(\omega;u)\right).u⟼(Ic​(u),Γϕ​(u),νosc​δνosc​​(u),Sϕϕ​(ω;u)). Here (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 α=dΦ−A,\alpha=d\Phi-A,α=dΦ−A, which is compared with the gauge-invariant superconducting phase difference ϕ=φ1−φ2−2πΦ0∫12A⋅dℓ.\phi=\varphi_1-\varphi_2-\frac{2\pi}{\Phi_0}\int_1^2 A\cdot d\ell .ϕ=φ1​−φ2​−Φ0​2π​∫12​A⋅dℓ. The earlier v0.5r2 version introduced a positive-control toy residual drive, RJ(u)=r0sin⁡2(πu),\mathcal R_J(u)=r_0\sin^2(\pi u),RJ​(u)=r0​sin2(πu), and used it as a morphology target for baseline-subtracted Josephson/SQUID observables. Version v0.6r2 extends this line by injecting synthetic noise and nuisance morphologies into mock observations, yk,iobs(A)=Aβkgi+nk,i+hk,i,y_{k,i}^{\rm obs}(A)=A\beta_k g_i+n_{k,i}+h_{k,i},yk,iobs​(A)=Aβk​gi​+nk,i​+hk,i​, where (g_i) is the centered residual morphology, (n_{k,i}) is a synthetic noise term, and (h_{k,i}) is a nuisance contribution. The central sensitivity outputs are the detection probability, Pdet(A)=1NMC∑m=1NMC1[trial m passes],P_{\rm det}(A)=\frac{1}{N_{\rm MC}}\sum_{m=1}^{N_{\rm MC}}\mathbf 1[\text{trial }m\text{ passes}],Pdet​(A)=NMC​1​∑m=1NMC​​1[trial m passes], the minimum detectable amplitude, Amin⁡=inf⁡{A:Pdet(A)≥0.95},A_{\min}=\inf\{A:P_{\rm det}(A)\ge0.95\},Amin​=inf{A:Pdet​(A)≥0.95}, and the false-positive rate, PFP=Pdet(0).P_{\rm FP}=P_{\rm det}(0).PFP​=Pdet​(0). Version v0.6r2 includes: 1. synthetic observation models for residual and nuisance channels;2. white-noise, high-noise, correlated-drift, heating-mimic, and mixed-nuisance scenarios;3. morphology-score and signal-to-noise recovery criteria;4. a multi-channel admission rule requiring at least two non-current channels to pass;5. matched-control subtraction for heating-like nuisance;6. false-positive and detection-probability curves;7. sensitivity tables and CSV outputs;8. an application-readiness ladder from toy positive control to device-screening protocol;9. laboratory deployment requirements for a future calibrated experiment;10. device-facing output interpretation tables. A key outcome of the synthetic study is that heating-like nuisance can overlap strongly with the residual morphology and can therefore create false-positive hazards if not controlled. The v0.6r2 update explicitly treats this issue by introducing matched-control subtraction and by distinguishing raw nuisance-contaminated recovery from controlled recovery. The application-readiness ladder used in this version is: - L0: toy positive-control morphology;- L1: synthetic-noise recovery and false-positive stress test;- L2: calibrated synthetic noise using measured device noise floors;- L3: laboratory pilot with real Josephson/SQUID data;- L4: device-screening protocol for phase-noise and barrier-quality diagnostics. The current v0.6r2 manuscript is positioned at L1. It is therefore a synthetic sensitivity and application-readiness note, not a calibrated laboratory demonstration. The included files provide the PDF manuscript, LaTeX source, figure-generation script, synthetic sensitivity summary CSV, detection-curve CSV, matched-control example CSV, simulation metadata JSON, and a source bundle for reproducibility. This work is a companion device-facing diagnostic note following the broader protocol record Prediction and Discriminating Experiments for the Residual Clock–Energy Framework, Version v0.4r1, DOI: 10.5281/zenodo.19846816.
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 EJ(u,t)=EJ(0)(u) [1+ζ(u,t)],E_J(u,t) = E_J^{(0)}(u) \, [1 + \zeta(u,t)],EJ​(u,t)=EJ(0)​(u)[1+ζ(u,t)], where ζ\zetaζ is a weak latent barrier perturbation. Feynman–Hellmann differentiation fixes the first-order frequency response, δln⁡ω01(u)=κJ(u) ζ(u)+O(ζ2),\delta \ln \omega_{01}(u) = \kappa_J(u) \, \zeta(u) + O(\zeta^2),δlnω01​(u)=κJ​(u)ζ(u)+O(ζ2), 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 ∣peven∣≤1.15×10−7,\vert{}p_{\mathrm{even}}\vert{} \le 1.15 \times 10^{-7},∣peven​∣≤1.15×10−7, it gives VIFeff=1.087,B∗=8.743×10−7,\mathrm{VIF}_{\mathrm{eff}} = 1.087, \qquad B_* = 8.743 \times 10^{-7},VIFeff​=1.087,B∗​=8.743×10−7, a nominal false-positive reserve of approximately 6.07%6.07\%6.07%, and a robust 95%95\%95% recovery floor amin⁡,95robust≃7.29×10−6.a_{\min,95}^{\mathrm{robust}} \simeq 7.29 \times 10^{-6}.amin,95robust​≃7.29×10−6. At the frozen half-flux operating point this corresponds to a synthetic frequency-response floor of about 15.5 kHz15.5\text{ kHz}15.5 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 xqpx_{\mathrm{qp}}xqp​. The same latent density drives both the qubit-frequency nuisance and a T1T_1T1​-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≤50 μs,t \le 50 \, \mu\mathrm{s},t≤50μs, 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)U(1)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⟶ρX⟶ωC(X),X \longrightarrow \rho_X \longrightarrow \omega_C(X),X⟶ρX​⟶ωC​(X), whereas the present work assumes the separate device-level chain ζ⟶EJ⟶ω01.\zeta \longrightarrow E_J \longrightarrow \omega_{01}.ζ⟶EJ​⟶ω01​. No microscopic derivation of X⟶ζX \longrightarrow \zetaX⟶ζ 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.
**Version v0.5 — Journal-Polish Update** This manuscript develops a conservative extension framework for Schrödinger dynamics in which an observed quantum system is embedded into an enlarged Hilbert space equipped with an auxiliary clock-information fiber. The fiber is not introduced as a macroscopic hidden spatial dimension, nor as a local hidden-variable register. Instead, it is treated as a quantum auxiliary sector carrying phase memory, clock/reference correlations, and correlation records that may appear as information loss or dephasing after partial tracing. The central contribution of this version is a residual-identifiability framework. Rather than claiming that unitary dilation itself is new, the paper asks whether a calibrated observed system can exhibit structured residuals after independent-bath subtraction and common-bath controls. The main diagnostic object is a log-residual statistic, Rχ(t;σref)=log⁡Cobs(t;σref)−log⁡Cind(t;σref),R_\chi(t;\sigma_{\rm ref})=\log C_{\rm obs}(t;\sigma_{\rm ref})-\log C_{\rm ind}(t;\sigma_{\rm ref}),Rχ​(t;σref​)=logCobs​(t;σref​)−logCind​(t;σref​), together with reference-covariance indices such as Iχ(t;σref)=∂σrefRχ(t;σref),\mathcal I_\chi(t;\sigma_{\rm ref})=\partial_{\sigma_{\rm ref}}R_\chi(t;\sigma_{\rm ref}),Iχ​(t;σref​)=∂σref​​Rχ​(t;σref​), and covariance/correlation-based diagnostics over controlled reference settings. The framework emphasizes that a nonzero residual alone is not sufficient evidence for an admissible clock-information fiber. A candidate residual must be non-factorizing, reproducible, reference-covariant, and not absorbed by independent local bath models, common classical noise, or generic common quantum bath controls. In particular, the admissible-fiber hypothesis is treated as a residual diagnostic condition, not as a proof of a new hidden variable, superluminal channel, or macroscopic extra dimension. The paper proves the elementary consistency statements for the enlarged dynamics: total unitarity under a self-adjoint total Hamiltonian, total energy conservation for time-independent generators, subsystem entropy growth under partial trace, and no-signalling compatibility under local trace-preserving operations. It also presents minimal toy models for phase-memory dephasing, shared-fiber cross-kernels, compact protected fiber sectors, and synthetic residual-fitting demonstrations. The synthetic fitting layer uses a representative model of the form $$C_{\rm obs}(t;\sigma_{\rm ref})=C_{\rm ind}(t)\exp[-a(\sigma_{\rm ref})t^2],\qquada(\sigma_{\rm ref})=a_0+a_1\sigma_{\rm ref}^2,$$ so that Rχ(t;σref)=−a(σref)t2.R_\chi(t;\sigma_{\rm ref})=-a(\sigma_{\rm ref})t^2.Rχ​(t;σref​)=−a(σref​)t2. This example is intended as an identifiability demonstration: after calibrated independent-bath subtraction, a candidate clock-information fiber signature should appear as a reproducible residual with controlled reference covariance, rather than as an arbitrary fitting term. The experimental outlook focuses on Ramsey/spin-echo baselines, dynamical-decoupling response, superconducting qubit and Josephson platforms, tunneling phase-delay diagnostics, and reference-modulation protocols. The manuscript also formulates a common-bath control envelope, ∣Rχ(t;σref)−Rctrl(t;σref)∣≤δR(t;σref),|R_\chi(t;\sigma_{\rm ref})-R_{\rm ctrl}(t;\sigma_{\rm ref})|\le\delta_R(t;\sigma_{\rm ref}),∣Rχ​(t;σref​)−Rctrl​(t;σref​)∣≤δR​(t;σref​), to express the requirement that a proposed fiber residual should not be absorbed by generic common-bath control models within statistical uncertainty. ### Main updates in v0.5 - Shortened journal-style title and abstract.- Removal of version-history language from the abstract.- Strengthened common-bath control criterion.- Addition of a formal common-bath control envelope.- Improved diagnostic separation between independent local baths, common classical noise, generic common quantum baths, and admissible clock-information fibers.- Algorithm-style synthetic residual-identifiability test.- Synthetic residual-fitting demonstration using reference-covariant dephasing.- Improved table formatting and manuscript presentation for journal-style readability. ### Scope and non-claims This work should be read as a structured framework for residual diagnostics in extended Schrödinger dynamics. It is not a hidden-variable theory, not a superluminal-signalling model, not a derivation of measurement collapse, and not a completed microscopic theory of mass, charge, or quantum gravity.
Version 4.0 public record for “A Nuisance-Aware Residual-Test Framework for Geometry-Induced Quantum Potentials in Curved Constrained Platforms”. Version 4.0 is a substantive revision of the public record. It updates the title and scientific framing, clarifies the established Jensen–Koppe–da Costa geometric-potential anchor versus the new nuisance-aware residual-test methodology, adds an explicit finite-width/disorder/spin-orbit/electrostatic validity envelope, strengthens the minimum experimental-design protocol, improves figure readability, and synchronizes the journal and Zenodo packages. The work is framed as a conservative methodological and design-level residual-test framework for geometry-induced quantum potentials in curved constrained platforms. The manuscript is anchored to the established Jensen–Koppe–da Costa geometry-induced potential for scalar particles constrained to smooth curves. The standard constrained-curve prediction is represented by the fixed coefficient −1/4, while the residual parameter delta-lambda is defined as a falsifiable null-test quantity. Standard constrained quantum mechanics predicts delta-lambda = 0. The manuscript emphasizes that any nonzero residual should first be treated as a platform anomaly unless it survives curvature scaling, nuisance projection, matched-reference controls, blind residual extraction, and cross-geometry replication. No experimental detection is claimed. The included simulations and prospective bounds remain design-level estimates. The package contains the manuscript, supplementary information, figures, reproducibility code, generated CSV/JSON data, metadata files, and public technical materials sufficient to regenerate the numerical validation tables, platform-specific energy-scale estimates, residual sensitivity plots, and figures.
I present a falsification-first experimental protocol for testing a proposed record-dependent contribution to residual dephasing in circuit-QED systems. The protocol is designed to distinguish a genuinely record-dependent residual from conventional apparatus-mediated effects introduced by manipulating record persistence itself. The protocol separates post-decoupling readiness controls from concurrent source-hypothesis tests; requires uncertainty-bounded contrasts between record-establishment conditions while controlling conventional physical channels; treats conventional explanations through calibration-compatible nuisance-model families where appropriate; and distinguishes nominal shot count from effective independent information under device, day, and miniblock correlations. It further requires operator discrimination, explicit conventional-noise accounting, preregistered decision rules, and independent device or architecture replication before any anomalous residual is eligible for escalation. A one-page laboratory handoff accompanies the paper to provide an implementation-oriented summary of the experimental decision path. Origin and relationship to prior work: This protocol was motivated by Francisco José Ruiz Izaguirre's phenomenological proposal concerning record-dependent residual dephasing (DOI: 10.5281/zenodo.22162999) and subsequent public technical exchange with the author. It is an independent falsification protocol, not an experimental confirmation of that proposal. No anomalous dephasing effect is claimed here.