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 the Odyssey enters Earth's atmosphere, alerting Jack with its sonic boom, it's daytime. When Jack arrives at the crash site, it's night time, and when he arrives back at Tower 49 with Julia's hibernation pod, it's daylight again.
When Jack is standing in the remains of the football stadium, he talks about the outcome of the last Super Bowl. But at the end, when he's listening to the recording of when his crew first met the Tet and he was talking to his commander, she was going to tell him about the game, but he told her not to because he wanted to watch it. He was then abducted by the Tet and never got to watch the game. So he had no idea how the game turned out.
When the three drones descend on the Scav base at the end of the movie, they sweep past the camera as they approach at full speed. The center drone nearest the cameras guns are partially outside and below their housings in this shot, incorrectly positioned by the VFX team.
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.
At night, a clear view of the exploded moon is visible in the night sky. With the debris caused by the explosion raining constantly into Earth's atmosphere, the night sky should be filled with "shooting stars" for the next several thousand years, especially in the deserted Earth's non-light-polluted evening skies.
Though its shown to be much further away, the sound of the exploding hydro-rig is heard approx 3 seconds after Jack sees the light of the explosion. This would put the large nuclear blast approx 3,200 feet (or 6/10th of a mile) away, not the many miles shown.
In the flashback scene when the Odyssey is approaching the alien vessel near Titan, the communication with "Mission" is immediate. In reality such a great distance would introduce a delay in both sending and receiving messages. It would take over one hour for a message to reach Earth.
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.
Julia looks still young after 60 years of sleep. She might be young if her spacecraft would have been traveling at the speed of light., but this spacecraft has been orbiting the Earth for 60 years, so she should be in her 80s.
When Julia first awakens from the Delta sleep, she is wearing a light white dress. Later, when she leaves the tower with Jack to recover the flight recorder, she's wearing the NASA flight suit and boots.
Shortly before the end of the film, Jack listens to the contents of the black box which he found in the crashed crew module with the hibernating "Odyssey" crew members. The recorded cockpit conversation between Victoria and Jack goes on after sealing off the module with other crew members and even continues after jettison of the module. At first glance it seems the cockpit conversation could no longer be on the black box, but the system could have been transmitting the recorded conversation to the crew module with the black box.
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 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.
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.
After the drone kills Victoria 49, Jack and Julia talk to "Sally", seeing her image on the monitor. Julia should have recognized the image as the Mission Control director from the original mission to Tet, and told Jack as much at that time.
The Tet was able to track Jack through a canyon by his DNA when the scavs captured him. If the Tet can scan DNA from orbit, why can it not scan the sleep module Jack is bringing up to the Tet with him?
When the Tet is destroyed, the attacking drones fall out of the air when they lose contact. However, the Tet loses contact with different areas of the earth as it orbits, meaning the drones would be useless many times during the day. However, they are shown fully functional and destroying the sleep pods after the Tet loses contact with Vicka and the tower, having went below the horizon.
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.