The implementation of the GRAVITY+ Adaptive Optics (GPAO) system at VLTI enables unprecedented sensitivity and stability in optical interferometry. This allows high-precision characterization of directly imaged exoplanets at medium spectral resolution, providing a new pathway for studying planetary atmospheres. We aim to quantify and characterize the short- and long-term stability of GRAVITY+ through a consecutive seven-hour observation of the bright and stable star β Pictoris, providing a benchmark for future exoplanet observations. We developed SIMTERFERE, a data-driven simulation tool that reproduces GRAVITY+ on-star observations using ancillary instrument and telemetry data. By comparing the simulations with the measured coherent fluxes, we traced the origins of systematic flux variations and assessed their impact on exoplanet contrast measurements. We find that the ~10% variations are dominated by throughput changes driven by variable fiber coupling, which depends on wavefront stability, atmospheric dispersion, and residual fiber offsets. These variations appear as smooth continuum changes across wavelength and can be effectively mitigated using second-order polynomial corrections. After removing these instrumental effects, the remaining ~1% variations are almost purely of telluric origin, which we can reliably correct down to the photon-noise limit (0.1% precision) using a contrast spectrum approach with linear airmass interpolation. The GRAVITY+ inferometric instrument is highly stable: low-order continuum and telluric variations can be corrected with high precision, making it uniquely capable of high-fidelity characterization of directly imaged exoplanets.
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