Combined, these give direct, model-based estimates of holonomy sensitivity (\beta) and holonomy variance susceptibility (\gamma) that can be reported and compared across platforms.
7. Summary --- falsifiable distinction
Equation (3) supplies a compact, falsifiable signature: oscillatory visibilities with a predictable period and an exponential envelope. Observation of this pattern (including phase drift and nonlocal partner response) would support the InterTop holonomy interpretation; robust absence of oscillatory structure within experimental sensitivity would falsify the class of InterTop models that attribute coherence modulation to synthetic holonomies in the tested parameter manifold.
C. Synthetic phase shifts under cavity parameter cycling
In InterTop the informational connection A\mathcal{A} produces measurable, deterministic phase shifts (synthetic phases) when control parameters are varied along open or closed paths in the experimental parameter space. In cavity setups these control knobs include the cavity leakage rate \kappa, emitter--cavity coupling gg, and detuning \Delta. This subsection derives the expected form of synthetic phase shifts under parameter cycling, shows how they follow from holonomy, and gives concrete experimental prescriptions to extract the holonomy coupling \beta and to discriminate this behavior from ordinary decoherence.
1. Holonomy origin of synthetic phase
Let p=(,g,,...)\mathbf{p}=(\kappa,g,\Delta,\dots) denote a vector of control parameters. In InterTop the mean synthetic phase accumulated between two informational nodes Ni,Nj\mathcal N_i,\mathcal N_j when the system is adiabatically transported in parameter space along a path C:p(s),s[0,1]\mathcal C:\mathbf{p}(s), s\in[0,1] is
C=C\mathbfcalA(p)dp,(C1)\Phi_{\mathcal C} \;=\; \alpha \int_{\mathcal C} \mathbfcal{A}(\mathbf{p})\cdot d\mathbf{p}, \tag{C1}
where \mathbfcalA(p)\mathbfcal{A}(\mathbf{p}) is the pullback of the informational connection onto the control-parameter manifold and \alpha is a dimensionful coupling constant connecting experimental units to informational phase. For an open path from p0\mathbf{p}_0 to p\mathbf{p} the relative phase between nodes is (p)(p0)\Phi(\mathbf{p})-\Phi(\mathbf{p}_0); for a closed loop \Gamma the holonomy is
holo()=\mathbfcalAdp=S()F,(C2)\phi_{\rm holo}(\Gamma)=\alpha\oint_\Gamma \mathbfcal{A}\cdot d\mathbf{p} \;=\; \alpha\int_{S(\Gamma)} \mathcal{F}, \tag{C2}
with curvature F=p\mathbfcalA\mathcal{F}=\nabla_{\mathbf p}\times\mathbfcal{A} and S()S(\Gamma) any surface bounded by \Gamma.
