Web-Halo Model Peak-Background Split (WHM-PBS): halo bias as a distribution, not a number

We present the Web--Halo Model Peak--Background Split (WHM-PBS), an analytic theory in which the large-scale bias of a dark-matter halo is inherited from its cosmic-web environment. Building on the Web--Halo Model, we use the Shen et al. moving barriers for ellipsoidal collapse generating the web hierarchy in which every halo sits inside a host filament, itself inside a sheet. Combined with the peak--background split, this picture replaces the deterministic bias--mass relation b(M h) with the bias of the host environment, averaged over the conditional mass function. As a result, halo bias b(M h) is no longer a number but a strongly skewed distribution. In this work we make use of this distribution in three different ways: as (i) a physically motivated prior on bias relations, (ii) a prediction on halo stochasticity, and (iii) a framework for assembly bias models. Regarding (i) we find that the density bias relations b 2(b 1) and b 3(b 1) stay tight, while the tidal bias b s squared(b 1) shows significant scatter, as found in N-body simulations. Regarding (ii), once including halo exclusion, our model reproduces the super- to sub-Poisson shot-noise trend of Baldauf et al. which we convert into a prior band on the EFT stochasticity amplitude parameters. Finally, regarding (iii) in the density sector it explains the bias--concentration--correlation inversion of Paranjape et al. at the characteristic mass (M h 1.7 times10 13,h-1), with no parameter tuned to assembly bias.