After Jack 49 gets the medical supplies from tower 52 he is surprised by Victoria 52. During their conversation Jack 49's rifle sling changes position multiple times covering up different portions of the 49 embroidered on his suit.
When Jack is investigating drone 172 in the library sinkhole: after the cable being snapped and him falling, you can see the sling on his weapon during the slow motion scene. But when the weapon falls off the ledge shortly after, the sling is not on the weapon.
When Jack first goes to the house by the river he puts a long play album on the stereo and plays the first track on one of the two sides. The song which starts is "Ramble On" by Led Zeppelin. This song first appeared on Led Zeppelin II, and subsequently on some Led Zeppelin compilation albums, but it has never been the first track on any side of a long play record.
When Jack 49 and 52 struggle for a weapon, it discharges and the blast strikes another character. Said character is shown to have a heavily bleeding wound as a result, similar to that which may occur with a ballistic weapon. However, all Tet-provided weapons are energy based and as such would cause a non-bleeding cauterized wound, if not a complete disintegration as is the case with larger vehicle-mounted weapons.
The design of the bubble ship resembles a helicopter including a tail rotor, however a tail rotor is provided to counteract the spin that would be caused by the main rotor in a helicopter. The bubble ship has no main rotor, so there would be no need for a tail rotor to counteract spin; however, a thruster on the end of the tail pylon would supply much more maneuverability than using the inboard engines to spin the ship.
Jack detects a Scav beacon signal emitted from the Empire State Building that is sending coordinates on Earth to the pre-invasion spacecraft Odyssey. The human survivors tell Jack that they decrypted GPS signals to use this transmission of coordinates. This wouldn't be possible that far in the future because the GPS satellites depend upon ground monitoring of their precise positions and slight changes to their orbits from ground controllers to maintain the GPS satellite coverage. With decades of no human intervention the GPS satellites would have long ago drifted, become out of position, transmitted incorrect signals, and failed from old age.
The Tet is established as a brilliant machine capable of strategic planning, yet it can't figure out why Jack wants to visit after being informed of the Scavs weaponizing power cells, the fact that a large number of them are missing, and with the assumption that Jack 49 has been affected by them. Or, if it does suspect Jack's intentions, it still lets him in without an escort. It is a tough sell to assume the Tet wants Jack's "cargo" so badly to risk such an obvious attack.
The moon is blown apart and it is said that this caused problems on Earth such as earthquakes. However the moon is still there in pieces and its mass and center of gravity would not have changed much. This would not have such an effect on Earth.
Sally (TET) would never have needed to go to all the trouble of cloning humans, giving them memories, erasing their memories, arming them, and forcing them to maintain drones when there were so many more drones and the capability to have recovery/repair drones in TET, or any of the water-processors. Clearly TET was able to defeat all the Earth's militaries and governments without human assistance - dealing with a few scattered scavengers would be easy.
When Jack 49 flies his bubble ship into zone 52, he crashes because his ship has been damaged by the attacking drone, not because "Sally" shut it down. That explains why he is later able to fly ship 52 back into zone 49 to deliver Victoria to the lake house without "Sally's" interference.
Jack 52 was left bound in the desert with no concept of what was going on, no vehicle, no communication with anyone other than TET, and no one knowing his location other than Jack 49. He could not have found the secret scav base to accompany them to the lake house.