A self-consistent orbital architecture for GG Tau A. I. Simultaneous orbital fitting of the hierarchical triple
GG Tau $A$ is a young triple system with the close pair, $Ab_1$--$Ab_2$, and $Aa$, surrounded by a massive circumtriple disk with a large inner cavity difficult to explain in a binary framework. We aim to determine the orbital architecture and individual stellar masses of GG Tau $A$ and assess how the available astrometric and disk-based constraints restrict the range of admissible solutions. We performed a joint fit using Oracle, developed specifically for hierarchical stellar systems. All astrometric measurements were placed in a common reference frame, since the historical wide-orbit astrometry is given relative to the unresolved photocenter of the $Ab$ subsystem. The fit included one new wide-orbit astrometric epoch and a prior on the total stellar mass derived from disk kinematics. We then applied, in post-processing, an additional geometrical constraint based on the observed center of the circumtriple disk. The fit yields orbital solutions compatible with the available astrometric and disk-based constraints and provides estimates of the individual stellar masses. The additional wide-orbit epoch only marginally reduces the range of admissible solutions. By contrast, the disk-center constraint leaves the favored orbital architectures largely unchanged but significantly tightens the stellar-mass partition. This yields posterior masses of $0.521^{+0.069}_{-0.051}$, $0.106^{+0.017}_{-0.013}$, and $0.79^{+0.10}_{-0.10}$ $M_\odot$ for $Ab_1$, $Ab_2$, and $Aa$, respectively. The reported values are posterior medians with 16th--84th percentile intervals. A joint treatment of the two orbital levels is required to recover a physically meaningful architecture and constrain the individual stellar masses. The resulting solutions provide a basis for future dynamical modeling of the circumtriple disk and for testing whether the observed cavity can further constrain the system architecture.