Below is a guide on all of the collectibles you can find during the Halo Infinite mission called The Tower.
Banished Audio Log – Escharum’s Testimony: 03 Harbinger
This audio log is located in a storage container at the base of The Tower on the eastern side.
Spartan Core #1
This core can be found in a storage crate under The Tower near where the giant hologram was during The Tower mission.
Mjolnir Armory – Midnight Griffin MKVII Armor Coating
Under the landing pad near the giant grab lift at The Tower.
UNSC Audio Logs – The Prisoner: 02 What was Inside
This audio log can be found leaning against a metal support beam on the second level after entering The Tower.
Spartan Core #2
This core can be found on the third level of The Tower near some UNSC crates.
Skull – IWHBYD
This skull can be found on the roof of The Tower. It is reachable by climbing up the outside of The Tower using the Grappleshot (with the Quickshot upgrade).