Stellar masses and ages in Gaia Data Release 4 from the Final Luminosity Age Mass Estimator algorithm

The masses and ages of stars are key quantities for understanding exoplanetary, stellar, and galactic evolution. In the context of Gaia, these parameters provide insights into the stellar populations, helping to trace the formation and history of the Galaxy. As part of the Gaia Data Processing and Analysis Consortium (DPAC), the Final Luminosity Age Mass Estimator (FLAME) pipeline processes Gaia data to derive stellar parameters comprising luminosities, radii, masses and ages. This paper discusses the methods and data used in FLAME for Gaia Data releases and the expected performances of FLAME for the 4th Gaia Data Release. FLAME comprises two main components: the first one, which is analytical, is used to estimate luminosity, radius, and radial velocity correction due to gravitational redshift by exploiting the atmospheric, astrometric, and photometric parameters produced within Gaia. The second is a model inference based on two main approaches: a classical minimization approach, and a Bayesian framework. It aims to derive mass, age, and evolutionary stage. The two step implementation offers flexibility in handling photometric properties that are prone to systematic errors. Tests with simulated data, the Sun, and well characterised samples of stars show that the methods in FLAME perform as expected, producing results in statistical agreement with the literature. We provide new stellar fundamental parameters for some high velocity stars, stars with very low mass companions, and a selection of stars in the Plato Field of View. In Gaia Data Release 4 approximately 500 million sources will have results from the pipeline. [abridged]