Perihelion Asymmetry in the Water Production Rate of the Interstellar Object 3I/ATLAS

3I/ATLAS is an interstellar object whose activity provides critical insights into its composition and origin. However, due to its orbital geometry, the object is too close to the Sun near perihelion to be observed from the ground, and space-based measurements are therefore required. Here we characterize the water production rate of 3I/ATLAS using SOHO/SWAN Lyman-$\alpha$ observations from 2025 November to December (heliocentric distances 1.4 to 2.2 au) with 3D Monte Carlo modeling. We report a peak post-perihelion water production rate of $Q_{\mathrm{H_2O}} \approx 4 \times 10^{28}$ molecules~s$^{-1}$, corresponding to a minimum active fraction of $\sim$30\% (assuming a maximum nucleus radius of 2.8 km). Comparison of our post-perihelion measurements with published pre-perihelion results reveals a heliocentric asymmetry, with an $r^{-5.9 \pm 0.8}$ scaling for the inbound rise, followed by a shallower $r^{-3.3 \pm 0.3}$ scaling during the outbound decline, where $r$ is heliocentric distance. The post-perihelion behavior indicates that the water production of 3I/ATLAS was driven primarily by the varying solar insolation acting on a stable active area. Combined with other evidence, including comparison with the hyperactive comet 103P/Hartley 2, our findings suggest that its water production is likely dominated by a distributed source of icy grains. Furthermore, it displayed remarkable stability in the activity with no signs of outbursts or rapid depletion of water production.