V(vs;T)=V0(vs)exp(12Varmot(vs)12Varslip(vs;T))cos((vs))(D1)\boxed{\; V(v_s;T) \;=\; V_0(v_s)\; \exp\!\Big(-\tfrac{1}{2}\,\mathrm{Var}_{\rm mot}(v_s) - \tfrac{1}{2}\,\mathrm{Var}_{\rm slip}(v_s;T)\Big)\; \big|\cos\!\big(\overline{\Phi}(v_s)\big)\big| \;} \tag{D1}
where:
V0(vs)V_0(v_s) is a slow (non-oscillatory) amplitude prefactor set by population imbalance and detector contrast (can be taken constant over a narrow vsv_s-window or included as a fitted smooth function),
(vs)\overline{\Phi}(v_s) is the mean synthetic holonomy (carrier phase),
Varmot(vs)\mathrm{Var}_{\rm mot}(v_s) is the variance from continuous vortex motion, and
Varslip(vs;T)\mathrm{Var}_{\rm slip}(v_s;T) is the variance accrued from discrete phase-slip events during measurement time TT.
Equation (D1) is the direct superfluid analog of the InterTop visibility model used for cavity systems (Sec. III.B--C), now written in terms of physically measurable vortex / tunneling quantities.
B. Parametric models for the components
Below we provide minimal, physically motivated parameterizations that experimentalists can fit to data. These forms are flexible --- they can be refined by microscopic modeling or simulation (sGPE) for a given platform.
a. Mean holonomy (carrier phase)
To leading order one expects a linear sensitivity of mean holonomy to superflow:
(vs)=0+vs+2vs2+.(D2)\overline{\Phi}(v_s) \;=\; \Phi_0 + \beta\,v_s + \beta_2 v_s^2 + \cdots. \tag{D2}
The dominant linear coefficient \beta is the primary parameter of interest (units: rad per velocity unit); 2\beta_2 captures curvature for larger drives.
b. Motion-induced variance
From Sec. IV.B (Eqs. (4)--(6)) the continuous- motion variance can be written in frequency-integral form; for practical fitting adopt either the overdamped thermal or resonant/inertial parametrizations:
(a) Overdamped / thermal regime (no strong inertial resonance)
Varmot(vs)G2kBTK(vs).(D3a)\mathrm{Var}_{\rm mot}(v_s) \;\simeq\; \frac{|\mathbf{G}|^2\,k_B T}{K(v_s)}. \tag{D3a}
Here K(vs)K(v_s) is an effective restoring stiffness (may depend on vsv_s via flow-induced softening), and G|\mathbf{G}| the geometric sensitivity of the loop.
(b) Resonant / inertia-sensitive regime (vortex mass mvm_v important)
Varmot(vs)CresG2S(0(vs))mv(vs)20(vs)2eff(vs),(D3b)\mathrm{Var}_{\rm mot}(v_s) \;\simeq\; C_{\rm res}\,\frac{|\mathbf{G}|^2\,S_\xi\big(\omega_0(v_s)\big)}{m_v(v_s)^2\,\omega_0(v_s)^2\,\Gamma_{\rm eff}(v_s)}, \tag{D3b}
with 0(vs)=K(vs)/mv(vs)\omega_0(v_s)=\sqrt{K(v_s)/m_v(v_s)}, eff/mv\Gamma_{\rm eff}\sim\eta/m_v the linewidth, S()S_\xi(\omega) bath spectrum, and CresC_{\rm res} an O(1)\mathcal O(1) prefactor from mode projection. This expression predicts strong, nonmonotonic sensitivity as control shifts 0\omega_0 across bath spectral features.
b. Slip-induced variance (phase-slip contribution)
If phase slips occur as Poisson events at rate (vs)\Gamma(v_s) with per-event phase-kick variance slip2\sigma_{\rm slip}^2 (averaged over spatial positions relative to the loop), then over measurement time TT
Varslip(vs;T)=(vs)Tslip2.(D4)\mathrm{Var}_{\rm slip}(v_s;T) \;=\; \Gamma(v_s)\;T\;\sigma_{\rm slip}^2. \tag{D4}
For quantum tunneling / instanton processes, take
(vs)0exp(Sinst(vs)),or (thermal)(vs)0(T)exp(E(vs)kBT).(D5)\Gamma(v_s) \;\simeq\; \Gamma_0 \exp\!\Big(-\frac{S_{\rm inst}(v_s)}{\hbar}\Big), \quad\text{or (thermal)}\quad \Gamma(v_s)\simeq \Gamma_0^{(T)}\exp\!\Big(-\frac{\Delta E(v_s)}{k_B T}\Big). \tag{D5}
Both forms predict strong, typically super-exponential sensitivity of slip-noise to control vsv_s.
c. Amplitude prefactor V0(vs)V_0(v_s)
Include a slow variation to account for population imbalance or detector response:
V0(vs)V(1+a1vs+a2vs2)1.(D6)V_0(v_s) \simeq V_* \big(1 + a_1 v_s + a_2 v_s^2\big)^{-1}. \tag{D6}
Often V0V_0 can be taken constant across a narrow scan.
C. Limiting regimes and distinctive signatures
From (D1)--(D6) we can extract the principal, falsifiable signatures:
1. Oscillatory visibility with reproducible nodes.
When Varmot+Varslip\mathrm{Var}_{\rm mot}+\mathrm{Var}_{\rm slip} is small enough to leave the carrier visible, the cos((vs))|\cos(\overline{\Phi}(v_s))| term produces oscillations and nodes at =2+n\overline{\Phi}=\tfrac\pi2 + n\pi. Decoherence-only models cannot produce parameter-tuned zeros with reproducible phases.
2. Nonmonotonic dependence due to inertial resonance.
In the resonant regime (D3b) visibility can dip sharply when 0(vs)\omega_0(v_s) crosses spectral peaks of the bath; this produces sharp resonant features (not simple monotonic decay).
3. Measurement-time dependence (slip-dominated regime).
If slip noise dominates, visibility depends explicitly on integration time TT through (D4). Varying TT will change the observed V(vs;T)V(v_s;T) in a predictable way --- a diagnostic not expected for time-local dephasing with a fixed rate independent of sampled event statistics.
4. Geometry dependence.
G|\mathbf{G}| encodes loop geometry; different loop areas yield different sensitivities (D3a/b,D4). Observing geometry-dependent suppression or event statistics supports holonomy interpretation.
5. Cross-correlation with vortex observables.
Direct imaging of vortices should correlate event rates (vs)\Gamma(v_s) with visibility dips and with non-Gaussian phase statistics.
D. Fitting strategy, parameter extraction, and falsification test
1. Data set: measure fringe visibility VV and phase ^\hat\Phi across a grid of vsv_s values and (when possible) integration times TT. Also measure (or infer) vortex resonance 0(vs)\omega_0(v_s), mv(vs)m_v(v_s), and bath spectrum S()S_\xi(\omega) where possible.
2. Step 1 --- phase (carrier) fit: fit ^(vs)\hat\Phi(v_s) to (D2) to estimate ,0\beta,\Phi_0. Test null =0\beta=0. Significant nonzero \beta is evidence for deterministic holonomy.
2. Step 2 --- visibility envelope fit: fit visibility amplitude (absolute value removed of cos|\cos| if phases known) to exp[12(Varmot+Varslip)]\exp[-\tfrac12(\mathrm{Var}_{\rm mot}+\mathrm{Var}_{\rm slip})] using parametric forms (D3a/D3b plus D4). Use independent measurements of mv,0m_v,\omega_0 to constrain D3b when present.
3. Model comparison: compare full InterTop model (D1--D6) against purely monotonic decoherence models Vdec(vs)=V0(d)exp[dec(vs)]V_{\rm dec}(v_s)=V_0^{(d)}\exp[-\Gamma_{\rm dec}(v_s)] using AIC/BIC and likelihood-ratio tests. Examine residuals for oscillatory power.
4. Time-scaling test: if slip term included, verify predicted linear-in-TT variance scaling by measuring V(vs;T)V(v_s;T) at multiple TT.
5. Geometry and imaging cross-checks: vary probe loop geometry and correlate with visibility sensitivity and with direct vortex imaging; confirm predicted dependence on G|\mathbf{G}| and on (vs)\Gamma(v_s).
Falsification rule (operational): within the experimental sensitivity region (characterized by reachable vsv_s span, SNR, and maximum TT), reject the class of InterTop models with nonzero \beta and slip-sensitive variance if:
