Cosmic web anisotropy is the primary indicator of halo assembly bias

The internal properties of dark matter haloes correlate with the large-scale halo clustering strength at fixed halo mass - an effect known as assembly bias - and are also strongly affected by the local, non-linear cosmic web. Characterizing a halo's local web environment by its tidal anisotropy alpha at scales approximately four times the halo radius, we demonstrate that these multiscale correlations represent two distinct statistical links: one between the internal property and alpha, and the other between alpha and large-scale (greater than or similar to 30 h(-1) Mpc) halo bias b(1). We focus on scalar internal properties of haloes related to formation time (concentration c(vir)), shape (mass ellipsoid asphericity c/a), velocity dispersion structure (velocity ellipsoid asphericity c(v)/a(v) and velocity anisotropy beta), and angular momentum (dimensionless spin lambda) in the mass range 8 x 10(11) less than or similar to M-vir/( h(-1)M(circle dot)) <= 5 x 10(14). Using conditional correlation coefficients and other detailed tests, we show that the joint distribution of alpha, b(1), and any of the internal properties c is an element of {beta, c(v)/a(v), c/a, c(vir), lambda} is consistent with p(alpha, b(1), c) similar or equal to p(alpha)p(b(1)vertical bar alpha)p(c vertical bar alpha), at all but the largest masses. Thus, the assembly bias trends c <-> b(1) reflect the two fundamental correlations c <->alpha and b(1)<->alpha. Our results are unaffected by the exclusion of haloes with recent major merger events or splashback objects, although the latter are distinguished by the fact that alpha does not explain their assembly bias trends. The overarching importance of alpha provides a new perspective on the nature of assembly bias of distinct haloes, with potential ramifications for incorporating realistic assembly bias effects into mock catalogues of future large-scale structure surveys and for detecting galaxy assembly bias.