In InterTop, this winding number is identified as a holonomy invariant:
Each vortex node Nv\mathcal{N}_v contributes a quantized holonomy of +2+2\pi.
Each antivortex node Nv\mathcal{N}_{\bar v} contributes 2-2\pi.
Thus, the appearance of vortex--antivortex pairs can be interpreted as holonomy creation events on the informational manifold.
4. Connection to visibility--distinguishability duality
In photon interferometry, visibility VV and distinguishability DD arise from superposition of path amplitudes. Analogously, in 2D superfluids:
Visibility: interference contrast of vortex density oscillations, reflecting coherent overlap of circulation holonomies.
Distinguishability: imbalance in vortex vs. antivortex populations (or asymmetry in amplitudes AvA_v vs. AvA_{\bar v}).
Thus, the duality relation V2+D21V^2 + D^2 \leq 1 is re-embedded in the vortex manifold: a balanced vortex--antivortex distribution maximizes coherence (high visibility), while asymmetry introduces distinguishability.
5. Experimental consequences
Observation of vortex-pair creation rates and their phase correlations provides a direct probe of holonomy creation.
Oscillatory vortex-density interference fringes are expected when tunneling is driven periodically, a signature not captured by simple mean-field superfluid theory but predicted by InterTop.
The quantized circulation invariants offer natural falsifiable markers for topological holonomies.
B. Variable vortex mass holonomy variance
In the InterTop picture the stability of a holonomy --- i.e. how sharply a synthetic phase is defined --- is controlled by the variance of the holonomy phase Var()\mathrm{Var}(\delta\Phi). The recent experimental observation that a vortex's effective mass is not constant but depends on its motion (and on the background flow / environment) gives a physically measurable mechanism that feeds directly into Var()\mathrm{Var}(\delta\Phi). Below we (i) set up a minimal dynamical model relating vortex motion to phase fluctuations, (ii) derive an explicit relation between vortex mechanical parameters (effective mass, damping, bath spectrum) and holonomy variance, (iii) show how that variance suppresses interference/visibility, and (iv) state concrete, falsifiable experimental consequences and measurement prescriptions.
1. Minimal dynamical model for a vortex and phase sensitivity
Model a single vortex core as a quasi-particle with coordinate X(t)\mathbf{X}(t) (2D), effective mass mvm_v, viscous damping \eta, in-plane restoring forces encoded by an effective stiffness tensor KK (from confinement or image interactions), driven by a stochastic bath force (t)\boldsymbol{\xi}(t). The Langevin equation is
