Similar to this question, but specifically regarding level 5 acquisition, how do you level up each field skill (Mechanical, Biology, Archaeology) to level 5? It is known that there are only 10 BLADE levels, and you can only reach level 4 in each field skill via BLADE leveling.
When you reach field skill level 4 in any skill, a mission called "Off the Record" will show up at the BLADE mission terminal. There is one for each field skill. Each quest has 3 riddles that you have to solve.
"Soar deep into the skies and seek the forgotten land's ultimate peak" - You need to fly (requires completion of chapter 9/"A Girls Wings") to Mount Edge Peak - it's the highest point in Oblivia and it's near a handful of floating islands.
"After dawn o'er the land of white, a rare red foe appears to smite" - You need to defeat a monster called Lyla's Galdr, which only appears between 5:00-7:00 AM. Some will suggest to go try to intercept this where it spawns, but it's much easier to kill it where it lands, since it will sit there for a period of time. First, go to FN 416. Near here is where it spawns (in the middle of all the other monsters). It will fly in and land in the middle. It does not spawn all the time, and it is a tyrant, so you'll notice if it's correct. If you do not see it spawn, head to the nearby time-changing station (there is one right nearby) and just set the time back to 5:00 AM and try again.
"Etchings of the ancient beasts to find? Seek a stony shore in fired climes." - You need to find collectibles called Savage Sketch. It appears in the regions around FN 514 and 515 - the east side of Cauldros.
After completing these 3 tasks, you will find an NPC named Solan hanging out in the alleys of the residential district in the evening. Talk to him and you get an upgrade to Mech 5.
"In the court of the forest gods, a brood of sentinels here do flock" - Discover Sentinels Nest, east of the Divine Roost in Noctilum
"Across the table where titans feast hides an insect you must defeat" - You have to defeat Calore (a level 62 tyrant). You can find it northwest of Titans Table, hanging out in a pit of lava.
"If you seek a pure white shell, to the east does it now dwell" - You have to collect White Forfex Shells from forfex monsters in Oblivia. There are some along the bottom coastline and a few other areas.
After completing this, head to Deliverance Park (residential district) in the early morning, and look for Solan on the outskirts of the park, across the street. He is not in the park, but he is nearby.
"Take this elevated corn to hand, atop the peaks of a delusional land" - You need to collect an item called Pale Maize. It spawns atop the delusian mountains in Sylvalum (both north and south peaks). I had better luck finding it during the day, but YMMV.
"Atop the forest's eternal cascade, a bitter baron awaits the blade" - You have to kill Acid Diluses. They're atop Everwhelm Falls, on the west side, in Noctilum. A well-placed Phoenix can kill the whole group at once.
"Above an early land do hover, courting birds whose seal you covet" - You have to collect Auravis Courtship Feathers from Auravis monsters. The easiest way I found to do this was to fast travel to the Leaning Ring in Oblivia, kill the two Auravis monsters atop the ring, then fast travel back to the Leaning Ring, respawning them.
Next, you have to find Solan in the Test Hangar at night time, talk to him, and get your upgrade.