This paper investigates the hierarchy of baryon physics assembly bias relations obtained from state-of-the-art hydrodynamic simulations with respect to the underlying cosmic web spanned by the dark matter field. Using the Bias Assignment Method we find that nonlocal bias plays a central role. We classify the cosmic web based on the invariants of the curvature tensor defined not only by the gravitational potential, but especially by the overdensity, as small-scale clustering becomes important in this context. First, the gas density bias relation can be directly mapped onto the dark matter density field to high precision exploiting the strong correlation between them. In a second step, the neutral hydrogen is mapped based on the dark matter and the gas density fields. Finally, the temperature is mapped based on the previous quantities. This permits us to statistically reconstruct the baryon properties within the same simulated volume finding percent precision in the two-point statistics and compatible results in the three-point statistics, in general within 1σ, with respect to the reference simulation (with 5-6 orders of magnitude less computing time). This paves the path to establish the best setup for the construction of mocks probing the intergalactic medium for the generation of such key ingredients in the statistical analysis of large forthcoming missions such as DESI, Euclid, J-PAS, and WEAVE.