5. Closed-loop (geometric) protocols
If two control parameters p,qp,q are available, adiabatic cycling around a closed loop \Gamma in the (p,q)(p,q) plane yields an accumulated synthetic geometric phase
geo()=(Apdp+Aqdq)=S()Fpqdpdq.(C10)\phi_{\rm geo}(\Gamma) \;=\; \alpha\oint_\Gamma \big(\mathcal{A}_p\,dp + \mathcal{A}_q\,dq\big) \;=\; \alpha\iint_{S(\Gamma)} \mathcal{F}_{pq}\,dp\,dq. \tag{C10}
Operationally:
Run the interferometer at (p0,q0)(p_0,q_0), measure start\Phi_{\rm start}.
Adiabatically vary (p,q)(p,q) around \Gamma back to (p0,q0)(p_0,q_0), measure end\Phi_{\rm end}.
The net accumulated phase =endstart\Delta\Phi=\Phi_{\rm end}-\Phi_{\rm start} should equal geo()\phi_{\rm geo}(\Gamma) if the process is adiabatic and holonomy-dominated.
Key diagnostic: geo()\phi_{\rm geo}(\Gamma) depends only on the loop geometry (area and curvature distribution) and not on traversal speed, within adiabatic limits. This property differentiates holonomic synthetic phases from dynamical phases and from classical drifts.
6. Control experiments and artifact rejection
To attribute observed phase shifts to informational holonomy (and not to trivial optical path changes, index changes, thermal drift, or electronics), the following controls are essential:
a. Single-arm phase check: verify that the noninteracting arm (or photon) shows no systematic phase drift while the interacting arm does. Nonlocal coincidence phase without single-arm drift strengthens the InterTop claim.
b. Classical phase injection: deliberately imprint a known classical phase on the optical path and confirm the measurement pipeline recovers it (calibration).
c. Switch-off test: repeat the protocol with cavity/emitter coupling turned off (no dark-state formation); \beta should drop to zero.
d. Engineered noise: increase holonomy variance (via controlled jitter of \kappa) and verify that phase linearity degrades into random jitter while visibility decays faster (consistent with e12Vare^{-\tfrac12\mathrm{Var}} suppression).
e. Loop reversibility/speed independence: test geometric phase invariance under loop speed changes and path deformations preserving area.
7. Statistical significance and falsifiability
Adopt the following statistical decision rule for \beta:
Null hypothesis H0H_0: =0\beta=0 (no synthetic holonomy). Alternative H1H_1: 0\beta\neq0.
Compute ^\hat\beta and its standard error ^\sigma_{\hat\beta}. Reject H0H_0 at significance level \alpha if ^/^>z1/2|\hat\beta|/\sigma_{\hat\beta} > z_{1-\alpha/2} (e.g., z0.99999975z_{0.9999997}\approx5 for a 5 claim).
Complement with model comparison (AIC/BIC) between InterTop joint model (phase + envelope) and best-fitting decoherence-only model (mono-tonic envelope, constant phase). A substantially lower AIC (AIC>10) for InterTop strengthens the inference.
Because phase estimates scale as (NV2)1/2\sigma_{\Phi}\gtrsim (N V^2)^{-1/2} for NN detected photons (Cramr--Rao bound), experimental design must ensure sufficient counts and visibility to achieve targeted ^\sigma_{\hat\beta}.
8. Illustrative numerical scaling (order-of-magnitude)
