Assembly bias from nuisance to probe I: the relation between galactic conformity and the linear matter clustering

Two-halo galactic conformity is commonly interpreted as a manifestation of galaxy assembly bias, but its statistical structure and physical origin remain unclear. We aim to write the quenched-neighbour statistic in correlation-function form, test whether its scale dependence follows the linear matter correlation function $\xi_{\rm mm}^{\rm lin}(r)$, and separate the contributions of halo-mass bias and assembly bias to its amplitude. Using galaxies in IllustrisTNG300-1 at $z=0$, we measure the two-halo galactic conformity statistic of quenched neighbours at distance $r$, $\Delta f_Q(r)$, and related quantities in real space, compute the required correlations, perform shuffling tests at fixed halo mass, compare several $\Delta f$ observables, and explore the transformed family $G_n$. We show explicitly that $\Delta f_Q(r)$ can be written directly in terms of correlation functions and that, over $\sim 2$-$40\,h^{-1}\,\mathrm{Mpc}$, it is well described by $A_{\rm fit}\,\xi_{\rm mm}^{\rm lin}(r)$. Thus nonlinear and baryonic terms do not dominate the residual scale dependence isolated by this statistic. Halo-mass bias alone predicts lower amplitudes than measured, while fixed-mass shuffling strongly suppresses the signal; in TNG300 the amplitude is therefore dominated by %fixed-mass assembly-dependent occupancy. galaxy assembly bias at fixed halo mass. Quenching, colour, and concentration share a common rescaled shape, whereas stellar-mass and halo-mass splits do not. These results suggest that galaxy assembly bias sets the amplitude of two-halo conformity, while the double-difference structure of the statistic suppresses nonlinear residuals when the compared populations have similar halo-mass and transition-scale structure.