6. Practical numeric scaling (order-of-magnitude) --- illustrative
As a rough guide, assume:
G1rad/m|\mathbf{G}|\sim 1\,\mathrm{rad/\mu m} (sensitivity of phase to core displacement),
thermal kBTk_B T equivalent energy scale ~ 102610^{-26}--102410^{-24} J depending on temperature,
effective stiffness K1012K\sim 10^{-12}--10910^{-9} N/m (strongly platform-dependent).
From Eq. (5) in overdamped limit:
Var()G2kBTK.\mathrm{Var}(\delta\Phi)\sim |\mathbf{G}|^2 \frac{k_B T}{K}.
If kBT/K103m2k_B T/K \sim 10^{-3}\,\mu\mathrm{m}^2, then Var103\mathrm{Var}\sim 10^{-3} and visibility suppression e12Var0.9995e^{-\tfrac12\mathrm{Var}}\sim 0.9995 (negligible). In contrast, near a resonance with strongly enhanced S(0)S_\xi(\omega_0), Var\mathrm{Var} can grow to 0.1\sim 0.1--1 producing measurable visibility suppression between 0.950.95 and 0.60.6. These numbers show the effect size is experimentally accessible with sensitive interferometry.
7. Theoretical caveats and robustness
The linearization GX\delta\Phi\approx\mathbf{G}\cdot\Delta\mathbf{X} holds for small core displacements relative to probe-path scale. For large displacements, nonlinear corrections (higher-order derivatives of \Phi) must be included.
Multi-vortex interactions: in realistic films multiple vortices interact; covariance becomes a many-body object Cov(Xi,Xj)\mathrm{Cov}(\mathbf{X}_i,\mathbf{X}_j). The same formalism extends but requires diagonalization into normal modes.
Nonthermal baths or non-Gaussian noise: Eq. (4) is general, but simple closed forms (5),(6) use Gaussian/thermal assumptions. For engineered baths the spectral density must be measured or modeled.
8. Summary --- conceptual and operational point
The observed variable vortex mass in 2D superfluids provides a concrete microscopic knob that controls holonomy variance in the InterTop picture. Through the chain vortex dynamics \Rightarrow position fluctuations \Rightarrow phase variance \Rightarrow visibility, experiments can quantitatively test whether coherence in the superfluid is best described as resonance in an informational topology. The key empirical prediction is a measurable, model-predictable mapping between vortex dynamical parameters (especially mvm_v and damping) and interferometric visibility; observation of the predicted dependencies (including resonant dips and inertial crossovers) would strongly support the InterTop hypothesis.
C. Tunneling events as phase slips and geometric transitions
Tunneling events in a 2D superfluid---spontaneous nucleation and annihilation of vortex--antivortex pairs under drive---are naturally interpreted in InterTop as discrete phase-slip events that effect topological transitions of the informational manifold. Below we (i) formalize tunneling as a phase-slip operator and an instanton process; (ii) relate event rates to an effective action and to holonomy changes; (iii) show how phase slips cause discrete jumps in synthetic phase and therefore observable interference; and (iv) derive experimentally testable predictions (statistics, spectra, and geometric-transition signatures) that distinguish InterTop from purely dissipative pictures.
1. Phase-slip operator and instanton picture
