Early in the film the flight recorder was retrieved from the crashed escape pod, Odyssey. Later at the end of the film, as Jack approaches TET, he listens to playback from the flight recorder. The playback indicates conversation with Victoria which runs all the way up to their point of entry into the TET. If Jack had released the escape pod, then the flight recorder, which was sealed inside, could not have recorded the conversation past the point of detachment from the Shuttle.
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.
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 Victoria and Jack are approaching the Tet, they converse with Sally at NASA as if they are in orbit around earth. In reality, a signal needs more than two hours to make a round trip from Saturn to Earth and back.
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.
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 Tet is shown to be capable of directly controlling all Earth-based vehicles and drones on several occasions, yet for some reason lets Jack 49 fly around for hours on end, shoot down drones, and travel to known Scav locations.