Geometric Confinement as a G-Only Baseline for Superconducting Phenomena: A Re-Entry Analysis of the QNFO Geometric Unification Framework

#1. Introduction

The QNFO corpus frames superconductivity within a "Geometric Unification Framework" (DOI 10.5281/zenodo.17074726) and a companion "Superconductivity Quadrangle" entry (DOI 10.5281/zenodo.18496889). The research idea re-entered here, "geometric confinement (G-only baseline)," asks a specific methodological question: if one strips the description down to geometry alone—confinement lengths, boundary shapes, and their ratio to intrinsic superconducting length scales—how much of the phenomenology of confined superconducting systems can be reproduced, and where does the baseline necessarily fail?

The motivation is methodological. Before attributing an effect to a microscopic mechanism, one should establish what a mechanism-free, geometry-only description predicts. In superconductivity this is pertinent because the literature itself is divided on how much of the phenomenology is universal. One line of work argues that whatever the mother normal states are, the superconducting condensate behaves in a BCS-like, "normal" fashion [1]; another line emphasises that band-structure diversity (non-parabolic bands, multi-band systems) is essential [3], [5]. A G-only baseline sits deliberately between these positions: it assumes nothing about pairing, but exploits the fact that geometry—thin films, disks, quasi-one-dimensional lattices, cavity boundaries—constrains the possible states of any condensate regardless of its microscopic origin.

This paper makes three contributions. First, it defines the G-only baseline precisely through a confinement parameter $\Gamma$ and derives its scaling relations (Section 3). Second, it computes concrete numerical values using explicitly stated assumed parameters and published qualitative constraints (Section 4). Third, it maps which published observations the baseline can organise—geometric barriers and flux domes in thin disks [8], two-tone mixing in superconducting quantum interference filters [7], coherence-time budgets of cavity-coupled qubits [6]—and which it cannot, namely critical temperatures and pairing-specific nonlinear relaxation [2], [3].

All quantitative content is either arithmetic performed in Section 4 from inputs stated there, or a projection under explicitly labelled assumptions with stated uncertainty bounds. We report no experimental measurements of our own and no simulation results.

#2. Background and Related Work

We discuss each bibliography entry in its exact supplied numbering, restricting every statement to what the entry's own title and summary text support.

Pairing universality versus material specificity. Reference [1] (arXiv:1603.03851v3, "Superconductivity driven by pairing of the coherent parts of the physical electrons") addresses the puzzle of how superconductivity in unconventional superconductors emerges from diverse mother normal states, and argues that whatever the mother normal states are, the superconductivity is normal, with BCS-like behaviours of the paired quasiparticles in condensation. This universality claim is the strongest support for a baseline approach: if the condensate side is universal, the residual variation among materials must live either in the mother normal states or in geometry. Reference [3] (arXiv:1811.11656v1) applies a dielectric function method for superconductivity to SrTiO$_3$, accounting for the non-parabolic dispersion of conduction-band charge carriers and for optical-phonon dispersion based on density functional theory calculations, and reports critical temperatures in agreement with experiments in the density regime (the supplied summary truncates at this point, so we cannot cite the specific density regime). This work is the clearest counterweight to a pure baseline: it demonstrates that band-structure detail—explicitly non-geometric—enters the prediction of $T_c$.

Confinement and dimensionality. Reference [4] (arXiv:1710.01668v1) studies superconducting properties of population-imbalanced fermionic mixtures in quasi-one-dimensional optical lattices, described by an attractive Hubbard model with a Zeeman magnetic field term, and investigates the ground-state phase diagram as a function of chemical potential and magnetic field (the summary truncates before the phase-diagram findings). The quasi-one-dimensional geometry is a geometric confinement in our sense; the population imbalance and Zeeman field are non-geometric inputs. We use this work only as evidence that confined, quasi-one-dimensional geometry is an active setting for superconducting phase structure. Reference [5] (arXiv:1712.06027v2) reports on Lifshitz transitions in multi-band Hubbard models for topological superconductivity in complex quantum matter, in the context of the Superstripes 2017 conference held in Ischia in June 2017, and poses the resistance of macroscopic quantum coherence to decoherence as a major challenge; the summary indicates a standard model of high-$T_c$ superconductivity for complex matter is under discussion but truncates mid-sentence, so no technical conclusion is available to us. Multi-band structure and Lifshitz transitions are material-side inputs that a G-only baseline excludes.

Microwave response and device physics. Reference [2] (arXiv:1608.06329v2) synchronously measured the second- and third-order nonlinear microwave response of a superconducting YBa$_2$Cu$_3$O$_7$ thin-film resonator using three input tones, a technique permitting local measurement and hence mapping of intermodulation distortion (IMD) inside the resonator, and found that second- and third-order IMD measured with a fixed probe relaxed in remarkably different ways after removal of a static magnet. This differential relaxation is material- and history-dependent; a G-only baseline has no memory variable, and we use it in Section 6 as a falsification probe. Reference [6] (arXiv:1409.3245v1) reports significant improvements in superconducting qubit coherence times achieved with three-dimensional microwave waveguide cavities coupled to transmon qubits, noting that while many measurements used superconducting aluminum cavities, other work involved qubits coupled to copper cavities with coherence times approaching $0.1\ \text{ms}$ (the summary truncates thereafter). This is a directly geometric observation: the three-dimensional cavity geometry supports long coherence times, and the $0.1\ \text{ms}$ figure is quoted from the entry, not derived by us. Reference [7] (arXiv:cond-mat/0608562v1) exploits the parabolic shape of the dc voltage output dip around $B=0$ of a Superconducting Quantum Interference Filter (SQIF) to mix weak external rf signals, detecting the two-tone response at the difference frequency $f_0 = f_1 - f_2$ for tones $f_1, f_2$ ranging from a few MHz up to $20\ \text{GHz}$. The parabolic dip is a geometric interference property of the loop array, and the stated frequency span is a bandwidth figure we quantify in Section 4.

Geometric barriers. Reference [8] (arXiv:0807.5129v1) analyses an ideal (no bulk pinning) flat type-II superconducting disk in a perpendicular applied field $H_a$: the first vortex nucleates at the rim when $H_a = H_0$, the threshold field, and moves quickly to the center; as $H_a$ increases above $H_0$, additional vortices join and produce a domelike field distribution of radius $b$, and the paper presents an analysis (summary truncated) of these geometrical barriers. This is the paradigmatic G-only system: with bulk pinning removed, the entire vortex phenomenology is governed by geometry. The entry supplies no numerical values for $H_0$ or $b$ and no functional form for $b(H_a)$, so our dome calculations in Section 4 are model projections, not reproductions of this work's numbers.

QNFO corpus. Reference [9] (QNFO: Superconductivity Quadrangle, DOI 10.5281/zenodo.18496889) and Reference [10] (QNFO: Geometric Unification Framework, DOI 10.5281/zenodo.17074726) are corpus entries whose supplied records contain only titles and identifiers and no summary text; we use them solely as the identifiers of the framework being re-entered and the source idea of this paper, and draw no substantive claims from them.

#3. Methods

#3.1 Definition of the G-only baseline

The G-only baseline retains exactly two classes of input:

1. Geometric data: a characteristic confinement length $L_g$ (disk radius $R$, film thickness $d$, lattice period, or cavity dimension) and the boundary shape.
2. Universal superconducting length scales: the penetration depth $\lambda_L$, which sets the screening response of any condensate, and the flux quantum $\Phi_0$, which sets the circulation quantum of any charged condensate.

It excludes: pairing interaction strengths, band dispersions, multi-band structure, population imbalance, pinning, and material-specific surface physics. Formally, for a candidate observable $O$ with material-dependent parameters $\mathcal{M}$, the baseline asserts

where $\mathcal{G}$ is the set of geometric parameters and $\epsilon$ the experimental tolerance. If $|\Delta_{\mathcal{M}}| \gt \epsilon$ is demonstrated, the observable carries material information.

#3.2 The confinement parameter

We define the dimensionless confinement parameter

where $d$ is the film thickness and $L_g$ the in-plane confinement length. The factor $2\lambda_L^2/d$ is the thin-film screening length $\Lambda$ that replaces $\lambda_L$ when $d \ll \lambda_L$; dividing by $L_g$ compares screening strength with confinement. Three regimes follow: $\Gamma \ll 1$ (screening dominates), $\Gamma \sim 1$ (screening and confinement compete; geometric barriers strongest), $\Gamma \gg 1$ (confinement dominates; barriers washed out).

#3.3 Model observables

We compute four families of G-only observables:

1. Vortex-entry threshold (disk-type, motivated by [8]). Equating the flux through the disk area to one flux quantum gives the minimal model threshold

This captures only the single-quantum nucleation constraint stated qualitatively in [8], not the full barrier analysis of that work; the true prefactor may differ.

1. Flux-dome growth (disk-type, motivated by [8]). We posit a confinement law for the mean induction inside the dome,

with $B^*$ the induction at which the dome reaches the rim ($b = R$) and $\gamma \gt 0$ a geometric exponent. Flux conservation (all applied flux confined within the dome) gives $\mu_0 H_a \pi R^2 = B_d \pi b^2$. Both $\gamma$ and $B^*$ are stated assumptions of the model, not measurements.

1. Algebraic mixing (SQIF-type, after [7]). The two-tone output frequency is $f_0 = f_1 - f_2$, fixed by the input tones and the mixing order; this is the trivially G-reproducible case.

1. Field-participation loss (cavity-type, motivated by [6]). For a cavity with a fraction $p_s$ of microwave field energy in the lossy surface region, the effective loss rate is $\kappa_{\text{eff}} = p_s\,\kappa_s + (1-p_s)\,\kappa_c$; we assume the field energy outside the confinement volume is a fraction $\eta$ of the inside energy, so $p_s = 1/(1+\eta)$.

#3.4 Constants and inputs

All arithmetic uses the following inputs, stated once. Flux quantum $\Phi_0 = 2.07\times10^{-15}\ \text{Wb}$ (standard constant). Penetration depth $\lambda_L = 100\ \text{nm} = 1.0\times10^{-7}\ \text{m}$ and film thickness $d = 50\ \text{nm} = 5.0\times10^{-8}\ \text{m}$ (illustrative assumptions). Disk radii $R = 1.0\ \mu\text{m}$ (threshold model) and $R = 100\ \mu\text{m} = 1.00\times10^{-4}\ \text{m}$ (dome model). Applied fields $\mu_0 H_a \in \{0.025, 0.050, 0.100\}\ \text{T}$. Confinement exponent $\gamma = 1$ and rim induction $B^* = 0.100\ \text{T}$ (assumed). Cavity parameters $\eta \in \{0.05, 0.20\}$ and $\kappa_s/\kappa_c = 100$ (assumed). Mixing tones $f_1 = 5.00\ \text{GHz}$, $f_2 = 4.99\ \text{GHz}$ (within the few-MHz-to-$20\ \text{GHz}$ range stated in [7]). Coherence time $\tau = 1.0\times10^{-4}\ \text{s}$ (from the $0.1\ \text{ms}$ stated in [6]) and assumed operating frequency $f_q = 5.0\times10^9\ \text{Hz}$.

#4. Analysis

Every input number is stated with its source; every arithmetic step is shown.

Derivation 1: screening length. With $\lambda_L = 1.0\times10^{-7}\ \text{m}$ and $d = 5.0\times10^{-8}\ \text{m}$,

Derivation 2: confinement parameter. With $L_g = R = 1.0\times10^{-6}\ \text{m}$,

A disk with these parameters sits in the intermediate regime where geometric barriers are expected to be significant, consistent with the qualitative picture of [8].

Derivation 3: vortex-entry threshold. With $R = 1.0\times10^{-6}\ \text{m}$ and $\Phi_0 = 2.07\times10^{-15}\ \text{Wb}$,

Since $H_0^{(G)} \propto R^{-2}$, a tenfold larger disk gives $H_0^{(G)}(10\ \mu\text{m}) = 6.589\times10^{-4}\times10^{-2}\ \text{T} = 6.589\times10^{-6}\ \text{T} \approx 6.59\ \mu\text{T}$: the threshold falls by two orders of magnitude per decade of radius.

Derivation 4: dome radius. Substituting $B_d = B^*(b/R)^{\gamma}$ into flux conservation:

With the assumed $\gamma = 1$, the exponent is $1/3$. At $\mu_0 H_a = 0.025\ \text{T}$: ratio $= 0.025/0.100 = 0.250$; $(0.250)^{1/3} = 0.62996$ (check: $0.62996^2 = 0.39685$, and $0.39685\times0.62996 = 0.25000 \approx 0.250$), so $b = 62.996\ \mu\text{m}$. At $0.050\ \text{T}$: $(0.500)^{1/3} = 0.79370$ (check: $0.79370^2 = 0.62996$, and $0.62996\times0.79370 = 0.50000$), so $b = 79.370\ \mu\text{m}$. At $0.100\ \text{T}$: $(1.000)^{1/3} = 1.000$, so $b = R$: the dome reaches the rim, consistent with the definition of $B^*$. Dome mean induction at $0.025\ \text{T}$: $B_d = B^*(b/R) = 0.100\times0.62996 = 0.062996\ \text{T}$; cross-check via flux conservation: $B_d = \mu_0 H_a (R/b)^2 = 0.025\times(1/0.62996)^2 = 0.025\times2.51984 = 0.062996\ \text{T}$. The two routes agree.

Derivation 5: vortex count. Assuming each vortex carries flux $\Phi_0$ and uniform dome induction,

independent of $\gamma$ by flux conservation. At $\mu_0 H_a = 0.025\ \text{T}$, $R = 1.00\times10^{-4}\ \text{m}$:

Check: $2.07\times10^{-15}\times3.7942\times10^{5} = 7.854\times10^{-10}$. Scaling linearly in $H_a$: $N \approx 7.588\times10^{5}$ at $0.050\ \text{T}$ and $N \approx 1.518\times10^{6}$ at $0.100\ \text{T}$.

Derivation 6: mixing frequency and bandwidth. With $f_1 = 5.00\times10^9\ \text{Hz}$ and $f_2 = 4.99\times10^9\ \text{Hz}$,

For the bandwidth, taking the span endpoints from [7] as $f_{\min} = 1.0\times10^6\ \text{Hz}$ (conservative reading of "a few MHz") and $f_{\max} = 2.0\times10^{10}\ \text{Hz}$,

Derivation 7: coherence phase-space product. With $\tau = 1.0\times10^{-4}\ \text{s}$ (from [6]) and assumed $f_q = 5.0\times10^9\ \text{Hz}$,

A qubit-cavity system with the coherence time reported in [6] sustains of order $10^6$ coherent cycles at a 5 GHz operating frequency, regardless of microscopic decoherence channels.

Derivation 8: cavity loss participation. With $\kappa_s/\kappa_c = 100$ and $p_s = 1/(1+\eta)$: for $\eta = 0.05$, $p_s = 1/1.05 = 0.95238$, giving $\kappa_{\text{eff}}/\kappa_c = 0.95238\times100 + 0.04762 = 95.286$. For $\eta = 0.20$, $p_s = 1/1.20 = 0.83333$, giving $\kappa_{\text{eff}}/\kappa_c = 83.333 + 0.16667 = 83.500$. Reference unconfined case ($p_s = 0.5$): $\kappa_{\text{eff}}/\kappa_c = 50.5$. Suppression ratios: $95.286/50.5 = 1.887$ and $83.500/50.5 = 1.653$ (check: $50.5\times1.887 = 95.29$; $50.5\times1.653 = 83.48$, rounding at the third digit).

Derivation 9: uncertainty propagation. If the disk radius is uncertain by a factor $r_{\pm}$ and $H_0^{(G)} \propto R^{-2}$, the threshold uncertainty is a factor $r_{\pm}^2$; a $\pm20\%$ radius uncertainty ($r_{\pm} = 1.2$) gives a threshold range $[0.659/1.44,\ 0.659\times1.44]\ \text{mT} = [0.458,\ 0.949]\ \text{mT}$. For the dome model, varying the exponent: $\gamma = 0$ gives exponent $1/2$ and $(0.250)^{1/2} = 0.5000$, so $b = 50.0\ \mu\text{m}$; $\gamma = 2$ gives exponent $1/4$ and $(0.250)^{1/4} = 0.7071$, so $b = 70.7\ \mu\text{m}$. The spread across $\gamma \in \{0,1,2\}$ at fixed field is $50.0$–$70.7\ \mu\text{m}$, roughly $\pm20\%$ about the $\gamma=1$ value; we carry this as the uncertainty band of the projection, while noting it holds $B^*$ fixed and is likely an underestimate.

#5. Results

All numbers below are computed in Section 4 or are labelled projections with stated assumptions.

1. Screening length (computed): $\Lambda = 400\ \text{nm}$ for the assumed $\lambda_L = 100\ \text{nm}$, $d = 50\ \text{nm}$.
2. Confinement parameter (computed): $\Gamma = 0.40$ for $R = 1.0\ \mu\text{m}$, placing the disk in the intermediate screening-confinement regime.
3. Vortex-entry threshold (model projection): $H_0^{(G)} = 6.589\times10^{-4}\ \text{T} \approx 0.659\ \text{mT}$ at $R = 1.0\ \mu\text{m}$, scaling as $R^{-2}$; at $R = 10\ \mu\text{m}$, $6.589\times10^{-6}\ \text{T} \approx 6.59\ \mu\text{T}$. With $\pm20\%$ radius uncertainty: $[0.458,\ 0.949]\ \text{mT}$. These are projections of the minimal single-quantum nucleation constraint, not measurements; the full barrier analysis of [8] would modify the prefactor.
4. Flux-dome growth (projection; assumptions $\gamma = 1$, $B^* = 0.100\ \text{T}$, $R = 100\ \mu\text{m}$):

Uncertainty band from exponent sensitivity: at $\mu_0 H_a = 0.025\ \text{T}$, $b \in [50.0,\ 70.7]\ \mu\text{m}$ for $\gamma \in [0, 2]$. The vortex count $N$ is independent of $\gamma$ and inherits only the uncertainty of the stated inputs $\Phi_0$, $R$, $H_a$.

1. Mixing (computed from the relation stated in [7]): for $f_1 = 5.00\ \text{GHz}$, $f_2 = 4.99\ \text{GHz}$, the output frequency is $f_0 = 10.0\ \text{MHz}$; the stated span from a few MHz to $20\ \text{GHz}$ corresponds to a ratio $2.0\times10^{4}$, i.e. $4.30$ orders of magnitude.
2. Coherence phase-space product (computed from stated inputs): $\omega\tau = 3.14\times10^{6}$ coherent cycles at $f_q = 5\ \text{GHz}$ with $\tau = 1.0\times10^{-4}\ \text{s}$ from [6].
3. Cavity loss participation (projection; assumptions $p_s = 1/(1+\eta)$, $\kappa_s/\kappa_c = 100$): $\kappa_{\text{eff}}/\kappa_c = 95.29$ at $\eta = 0.05$ and $83.50$ at $\eta = 0.20$, versus $50.5$ unconfined; confinement lowers the loss rate by factors $1.89$ and $1.65$ respectively in this model.
4. Scope result (structural): the G-only observable set $\mathcal{O}_G = \{H_0^{(G)},\ \Gamma\text{-regime},\ b(H_a),\ N,\ \text{bandwidth ratios},\ \omega\tau\}$ contains geometry-dominated quantities only; $T_c$ and material-specific nonlinear relaxation lie outside it.

#6. Discussion

Limitations. The baseline is deliberately mechanism-free, and this is both its strength and its ceiling. First, it cannot predict $T_c$: as the SrTiO$_3$ dielectric-function analysis shows [3], critical temperatures depend on non-parabolic band dispersion and phonon dispersion, inputs the baseline excludes by construction. Second, the minimal threshold model $H_0^{(G)} = \Phi_0/(\pi R^2)$ ignores the full geometrical-barrier physics analysed in [8], where the first vortex nucleates at the rim and subsequent vortices build a domelike distribution of radius $b$; our model captures only the nucleation scale, and the true threshold prefactor may differ. The dome model of Result 4 is a two-parameter fit ($\gamma$, $B^*$) whose parameters we assumed, not derived from [8] or measured; the supplied summary of [8] gives no functional form for $b(H_a)$, so our power law is one admissible ansatz among many, and the $\pm20\%$ band almost certainly underestimates the true model uncertainty. Third, the assumed parameters ($\lambda_L = 100\ \text{nm}$, $d = 50\ \text{nm}$, $R$, $f_q = 5\ \text{GHz}$) are illustrative; the computed values of $\Gamma$, $H_0^{(G)}$, and $\omega\tau$ scale with these assumptions and should be read as normalisations, not predictions for any specific material system. Fourth, the cavity participation model assumes a single lossy surface region and a fixed ratio $\kappa_s/\kappa_c = 100$; the summary of [6] attributes coherence improvements to reduced surface loss in three-dimensional cavities without giving loss ratios, so the factor-$1.89$ suppression is a model output, not a comparison with any measured value.

Failure modes and falsification. (i) If geometric observables such as vortex-entry thresholds were found to depend strongly on material substitution at fixed geometry, the claim that geometry dominates $\mathcal{O}_G$ would fail. (ii) The differential relaxation of second- and third-order IMD after removal of a static magnet, observed in the YBa$_2$Cu$_3$O$_7$ resonator of [2], shows two nonlinear orders relaxing "in remarkably different ways" in the same geometry; a purely geometric instantaneous model has no memory variable and cannot produce this, making it the strongest candidate for an observable that resists G-only description. (iii) If the universality claim of [1]—BCS-like condensate behaviour independent of mother normal state—were violated, the clean separation between universal condensate physics and material-specific normal-state physics would collapse. (iv) If coherence times of cavity-coupled qubits were found to be independent of cavity geometry at fixed materials, the geometric reading of the improvements reported in [6] would be weakened, though the $0.1\ \text{ms}$ figure quoted there would then need a non-geometric explanation. (v) The dome model itself is falsifiable: measuring $b(H_a)$ on a pinning-free disk and finding an exponent inconsistent with $b/R \propto H_a^{1/3}$ (or any value in the $\gamma \in [0,2]$ band of Derivation 9) would refute the assumed confinement law.

Open questions. Whether a sharper G-only threshold can be derived from the full geometrical-barrier analysis of [8] without material input; whether the $\omega\tau$ product is a useful figure of merit across qubit platforms or merely restates the coherence time; and whether the IMD memory effect of [2] can be captured by adding a single scalar history variable to the baseline, which would be the minimal extension beyond G-only.

#7. Conclusion

We have constructed a G-only baseline for confined superconducting systems, retaining only geometric lengths, the penetration depth $\lambda_L$, and the flux quantum $\Phi_0$. With fully shown arithmetic under stated assumptions we computed a screening length $\Lambda = 400\ \text{nm}$, a confinement parameter $\Gamma = 0.40$ for a representative disk, a model vortex-entry threshold $H_0^{(G)} = 6.589\times10^{-4}\ \text{T}$ at $R = 1.0\ \mu\text{m}$ scaling as $R^{-2}$, a flux-dome radius $b = 62.996\ \mu\text{m}$ at $\mu_0 H_a = 0.025\ \text{T}$ for a $100\ \mu\text{m}$ disk with vortex count $N \approx 3.794\times10^{5}$, a mixing bandwidth ratio of $2.0\times10^{4}$, a coherence phase-space product $\omega\tau = 3.14\times10^{6}$, and model cavity-loss suppression factors of $1.89$ and $1.65$. The baseline organises geometry-dominated observables, including those of the pinning-free disk of [8], the SQIF mixer of [7], and the cavity-qubit systems of [6], but cannot predict material-dependent quantities such as $T_c$ [3] or history-dependent nonlinear relaxation [2]. Its value is methodological: it fixes what any microscopic mechanism must explain on top of geometry, and it states falsification conditions under which the geometric description itself would fail.

#References

[1] Superconductivity driven by pairing of the coherent parts of the physical electrons. arXiv:1603.03851v3. https://arxiv.org/abs/1603.03851v3 [2] Relaxation of Microwave Nonlinearity in a Cuprate Superconducting Resonator. arXiv:1608.06329v2. https://arxiv.org/abs/1608.06329v2 [3] Superconductivity in SrTiO$_{3}$: dielectric function method for non-parabolic bands. arXiv:1811.11656v1. https://arxiv.org/abs/1811.11656v1 [4] Phase transitions in quasi-one dimensional system with unconventional superconductivity. arXiv:1710.01668v1. https://arxiv.org/abs/1710.01668v1 [5] Lifshitz transitions in multi-band Hubbard models for topological superconductivity in complex quantum matter. arXiv:1712.06027v2. https://arxiv.org/abs/1712.06027v2 [6] Copper waveguide cavities with reduced surface loss for coupling to superconducting qubits. arXiv:1409.3245v1. https://arxiv.org/abs/1409.3245v1 [7] Two tone response in Superconducting Quantum Interference Filters. arXiv:cond-mat/0608562v1. https://arxiv.org/abs/cond-mat/0608562v1 [8] Geometrical barriers and the growth of flux domes in thin ideal superconducting disks. arXiv:0807.5129v1. https://arxiv.org/abs/0807.5129v1 [9] DOI 10.5281/zenodo.18496889. QNFO: Superconductivity Quadrangle. [10] DOI 10.5281/zenodo.17074726. QNFO: Geometric Unification Framework.

#Appendix A. Divergence report

Three independent drafts (A, B, C) were written from the same input block. The following divergences were identified and resolved by explicit convention choice in the main text.

D1: Bandwidth lower endpoint of [7]. Draft A read "a few MHz" conservatively as $f_{\min} = 1.0\times10^{6}\ \text{Hz}$, giving a span ratio $2.0\times10^{4}$; Draft B read it as $3.0\times10^{6}\ \text{Hz}$, giving $6.7\times10^{3}$. The disagreement is purely a reading convention for a qualitative phrase in the supplied summary. Convention adopted: the conservative $f_{\min} = 1.0\times10^{6}\ \text{Hz}$ (Result 5), with the caveat that the true ratio is smaller if "a few" exceeds one.

D2: Dome exponent $\gamma$. Draft A assumed $\gamma = 1$; Draft C assumed $\gamma = 2$, citing a steeper confinement ansatz. No draft derived $\gamma$ from [8], whose supplied summary gives no functional form for $b(H_a)$. Convention adopted: $\gamma = 1$ in the main text, with the sensitivity band $b \in [50.0,\ 70.7]\ \mu\text{m}$ for $\gamma \in [0,2]$ carried as the projection uncertainty (Derivation 9).

D3: Cavity loss convention. Drafts A and B used the participation $p_s = 1/(1+\eta)$; Draft C used $p_s = \eta/(1+\eta)$, interchanging the roles of surface and bulk. The two conventions give reciprocal suppression factors. Convention adopted: $p_s = 1/(1+\eta)$, so that small $\eta$ (field mostly in the surface region) yields high loss, matching the qualitative direction of the improvements reported in [6].

No divergence was found on the computed values of $\Lambda$, $\Gamma$, $H_0^{(G)}$, $N$, $f_0$, or $\omega\tau$; all drafts reproduced the same arithmetic from the same stated inputs.

#Appendix B. Claim attribution