These puzzle sequences in the main quest Best Left Forgotten has rather unique gameplay and mechanics that they deserve having individual pages on their own for solutions. There are five memory sequences that you have to unlock, each adding new objectives in the quest and marks off additional locations for you to check out. Before you start off, here are the basics.
You'll have to make a path for the Indexers, the little friendly bug-like critters , to reach the yellow data stream and protect them from the system's automatic defenses as they carry the data back to your “base” or your blue data stream. If you fall out of the platform, you'll respawn in your starting position and have the option to leave the simulation or not. Whatever you do in the simulation is permanent even after you leave. In this simulation, you can only move blocks and place defenses. You cannot attack the firewall or the countermeasures directly.
There are natural hindrances, red walls act as firewalls that you need to take down first before your Indexers can move. By default, there are Decoder Beams constantly firing anti-firewall beams and you'll have to guide the beam through proper placement of Decoder Relays (blocks that bend the beam in any four directions of the same level). Firewalls can only be taken down if the decoder beam hits their weak spot, which is obviously the only different-colored cube in the firewall.
Once you successfully take down the firewalls, you can lay down Code Blocks (generic blocks) to fill the gaps and let your Indexers to cross. Later on, you'll need these blocks to create your own bridges to reach far off platforms. The system's automatic security will trigger as soon as you bring down the last firewall and proper placement of Defense Constructs or auto-turrets along your indexers' path is necessary to protect them from the red floating “Sentry Intrusion Countermeasures”.
Moving and placing these components are similar to how you build and move stuff using your workshop menu.
Code Blocks are the generic blocks that can be relocated and stored. They are mainly used to fill the gaps on the floor and bridge paths for your Indexers to cross. It's advisable to collect and store them whenever you find them, especially those placed inconveniently or piled up in unhelpful positions.
Decoder Relays are the blocks that bend the Decoder Beam to any four directions. They can be rotated but they can't bend the beam up or down.
Defense Contructs are automatic turrets that shoot down the countermeasures or red balls that will attempt to “erase” your Indexers. They have a significant long range and can chain their attacks, allowing them to focus fire even when one turret detects and shoots an enemy from afar.
Finally, we have the Indexers. These critters will move on their own and you have to protect them after successfully breaking the firewalls. If they are destroyed, a new Indexer takes its place and makes the journey again from your data stream to the yellow data stream until it can successfully take a memory piece and return it back to base. You'll have five Indexers at once, each capable of carrying 20% of memory at a time. That said, it's possible to get all the memory pieces in one go as long as no Indexers get destroyed during their journey.