While LFS has provided an approach to build a Linux system entirely from source code, it builds the LFS system in a chroot environment. The chroot environment is almost seperate from the host distribution, but it still relies on the hardware and Linux kernel from the host system.
The Cross Edition contains modification to the LFS book, allowing to boot the temporary system instead of chroot. So it's possible to bootstrap a Linux system for a target where no Linux distribution exists. And, the kernel ABI may frequently change at the early stage porting Linux to a new hardware. The modification in this edition completely removes the dependency on the kernel from the host distribution, allowing to rebuild the entire system with a new kernel ABI.
If you already have a Linux distribution running on the hardware where you need a Linux From Scratch system, with the kernel ABI you want, and it's compatible with the Host System Requirements specified by the LFS book, it's likely you should read the original LFS book instead of this edition.
This edition contains instructions to bootstrap a Linux system for x86 (32-bit or 64-bit). It may look irrational because it's easy to find a host distribution for x86, but this edition can be a start point where you can further modify the instruction for other hardware.
A project named Cross Linux From Scratch was forked from LFS in 2006, by Joe Ciccone, Jim Gifford, and Ryan Oliver. It has been inactive since 2017. Its history can still be explored at https://trac.clfs.org/.
In 2020, Pierre Labastie modified the LFS book to cross-compile the temporary packages before entering the chroot environment. The main purpose of this approach was to fix multiple issues found by Iterative Comparision Analysis. With Pierre's work, the only change needed by this edition is basically “to reboot instead of chroot.” However, some additional steps is needed to make the system bootable at the end of Chapter 6, and setting up an environment suitable for building and test packages at the start of Chapter 7. Then this Cross Edition is created to demonstrate these steps.
Unlike the legacy CLFS project, in the Cross Edition we don't build any package containing an init process for the temporary system to boot on the target machine. The temporary system is booted with a shell running as PID 1, and some necessary jobs of a normal init process are manually done with commands. This minimizes the difference from the main LFS book, and it's also helpful for learning the initialization steps booting a Linux system.