  We present the GeneRalized ANd Differentiable Halo Occupation Distribution
(GRAND-HOD) routine that generalizes the standard 5 parameter halo occupation
distribution model (HOD) with various halo-scale physics and assembly bias. We
describe the methodology of 4 different generalizations: satellite distribution
generalization, velocity bias, closest approach distance generalization, and
assembly bias. We showcase the signatures of these generalizations in the
2-point correlation function (2PCF) and the squeezed 3-point correlation
function (squeezed 3PCF). We identify generalized HOD prescriptions that are
nearly degenerate in the projected 2PCF and demonstrate that these degeneracies
are broken in the redshift-space anisotropic 2PCF and the squeezed 3PCF. We
also discuss the possibility of identifying degeneracies in the anisotropic
2PCF and further demonstrate the extra constraining power of the squeezed 3PCF
on galaxy-halo connection models. We find that within our current HOD
framework, the anisotropic 2PCF can predict the squeezed 3PCF better than its
statistical error. This implies that a discordant squeezed 3PCF measurement
could falsify the particular HOD model space. Alternatively, it is possible
that further generalizations of the HOD model would open opportunities for the
squeezed 3PCF to provide novel parameter measurements. The GRAND-HOD Python
package is publicly available at https://github.com/SandyYuan/GRAND-HOD .
