A vortex--antivortex tunneling event at spacetime point (r0,t0)(\mathbf{r}_0,t_0) inserts a quantized 22\pi phase winding on any loop encircling r0\mathbf{r}_0. We model the effect on the informational wavefunctional [M]\Psi[\mathcal{M}] by a unitary (or in a stochastic description, jump) operator S^r0\hat{S}_{\mathbf r_0} that shifts the holonomy on loops \gamma enclosing r0\mathbf r_0:
S^r0:+2q(,r0),\hat{S}_{\mathbf r_0}:\quad \Phi_\gamma \mapsto \Phi_\gamma + 2\pi\,q(\gamma,\mathbf r_0),
where q(,r0){0,1}q(\gamma,\mathbf r_0)\in\{0,\pm1\} is one if \gamma winds once about r0\mathbf r_0 (sign depending on vortex vs antivortex). In a path-integral (instanton) formalism, tunneling corresponds to nontrivial saddlepoint configurations inst(r,t)\varphi_{\rm inst}(\mathbf r,t) of the action S[]S[\varphi] with finite action SinstS_{\rm inst}. The semiclassical rate per unit area per unit time is
tunnAeSinst/,\Gamma_{\rm tunn} \;\approx\; \mathcal{A}\,e^{-S_{\rm inst}/\hbar},
with prefactor A\mathcal{A} set by fluctuation determinants. In driven or finite-temperature settings this Arrhenius-like form is modified (e.g. Kramers form, or Schwinger-like exponentials), but the key point is that tunneling is exponentially suppressed unless control parameters (superflow vsv_s, local potential, or drive amplitude) reduce the effective barrier.
2. Holonomy change and geometric transition
A phase-slip is therefore a discrete change in the holonomy field ()\Phi(\gamma). For an observer measuring an interference loop \gamma, a single vortex creation--annihilation sequence that crosses the loop produces a net holonomy increment =2\Delta\Phi=\pm 2\pi. In InterTop language this is a quantized geometric transition---the informational manifold has changed its topological charge content.
For an ensemble of loops or a continuously monitored loop, the time-dependent holonomy is
(t)=(0)+2nqn(ttn),\Phi_\gamma(t) \;=\; \Phi_\gamma(0) + 2\pi\sum_{n} q_n \Theta(t - t_n),
where tnt_n are event times and qnq_n the charges. When events are rare, (t)\Phi_\gamma(t) shows discrete steps (phase slips); when events are frequent, (t)\Phi_\gamma(t) becomes a stochastic process with continuous-like drift and large variance.
3. Effect on probes: discrete jumps, telegraph noise, and visibility collapse
