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Errors in geography
At around 37:00 right after a scene showing a dialogue between Cade Yeager and Izabella, it goes to a scene in Africa and the location is written in the bottom left corner as Namibia, West Africa. That is a horrendous geographical error akin to saying that New York is on the west coast of U.S.A. when in actual fact it is on the east. Namibia is in Southern Africa and NOT in West Africa. And that is a profoundly huge distinction as far as geography is concerned.
(at around 1h 3 mins) When Vivian is kidnapped by Hot Rod, she is wearing heels. When Hot Rod spits her out of the car at Burton's castle, she is wearing tennis shoes.
In Transformers: Dark of the Moon (2011) we can clearly see that Cybertron is much larger than Earth but in this movie it is only about a third or half the size of Earth.
(at around 1h 12 mins) A statue of Optimus from ancient history is shown in Sir Edmund's office. The design used for Optimus here is clearly his knight armor that he wears in this film, despite the fact that he did not look like this until the fourth installment.
The Battle of Chicago occurred in Transformers: Dark of the Moon (2011). In Transformers: Age of Extinction (2014) Chicago had been restored as an active city, but in this movie it is back to a post battle wasteland.
(at around 48 mins) When the Autobots are leaving the junkyard before the Decepticons arrive we see Squeaks with both his original arms. He had one shot off back in Chicago and Izabella attaches a Decepticon arm to him.
(at around 1h 35 mins) While the submersibles are following the submarine into the alien space ship the Navy Seal says "we are about to lose radio contact".... radio waves cannot pass though water therefore radios are not used for under water communication.
The gravity of an approaching "planet-sized object" (namely Cybertron) would effect Earth, most notably massive tidal changes. However, the only changes seen are from physical impacts.
Throughout the movie military characters refer to alien attack aircraft as "bogeys". This is common in Hollywood movies but is wrong: a "bogey" is an unidentified aircraft, i.e. usually an aircraft caught on radar that has yet been identified as friend or foe. An enemy aircraft is called a "bandit".
(at around 28 mins) When first seen in the junkyard, Samurai Autobot "Drift" is balancing on his sword on the roof of a car, causing no damage apart from a few scrapes. The sword alone would have been heavy enough to at least dent the roof; adding the weight of Drift would have caused him to skewer the car like a kebab.
(at around 2h 5 mins) Towards the end of the movie Colonel Lennox states that the Osprey's flight ceiling is 12,000 ft and the target where the battle will be taking place is at 21,000 ft, "nearly double" and the air would be too thin for the Osprey's rotors to "bite". This is incorrect. The Osprey's operational flight ceiling is in fact 25,000 ft, well within the operational area.
(at around 45 mins) When Daytrader brings a new voice box for Bumblebee, he lies on his back like a patient at the dentist while Cade and Izabella fix it for him. In several shots, Bumblebee's head is clearly a prop as the rest of his body has disappeared behind Cade and Izabella.
(at around 1h 27 mins) At one point during the car chase at Buckingham Palace, bystanders are shown watching and/or filming the scene. Likewise, the barriers blocking the streets and sidewalks for filming are visible.
(at around 5 mins) Merlin (Stanley Tucci) is given a powerful staff . Later a horseman is seen with it but it's not Tucci.
(at around 3 mins) Merlin drinks out of a glass bottle with a grooved neck that is clearly for a screw-on cap. That's at least a few hundred years ahead of its time.
(at around 1h 15 mins) In the flashback scene when Sir Edmund tells about Arthur and his knights working together with the Transformers, the environment where the Round Table is placed is shown as a Gothic style cathedral ruin. In reality Gothic style didn't appear until the 12th century, so cc. 700 years later. The clothing also show similar anachronisms.
(at around 1h 21 mins) When Vivian comes home with Cade, the first shot where the mirror is visible in the back shows the vague cameraman bending over his own camera.
(at around 2h 10 mins) A cameraman can be seen right behind Cade and other characters as the camera pans up towards the ignition chamber.
(at around 9 mins) The camera pans past a wrecked subway and its reflection can be briefly seen on the broken glass plate resting aside the wreckage.
During the Bumblebee and Barricade chase sequence, Cogman blasts a Lexus police car and within a few frames, you can momentarily see the camera car's shadow in two different shots.
The "Trinity Library" shown is not in London; it is in Trinity College Dublin, in Ireland.
(at around 1h 40 mins) The ESA scientist says that "the planet hid its approach behind our star", however the shot of Cybertron immediately after shows it is moving from outer space towards Earth and the Sun.
(at around 1h 29 mins) The Royal Navy's Submarine Museum is actually nowhere near London as the movie depicts; it is actually in Gosport, near Portsmouth, in Hampshire.
(at around 1h 22 mins) Hopkins is shown running down St Helens Passage in Oxford to get to the library. Not only does that mean he's running away from the library, he's also going down a dead-end that leads to only one thing - a pub (the sign for which is clearly visible).
(at around 2h) Stonehenge as shown is not as isolated in open countryside and from main roads as portrayed; the A303 among other roads runs past the monument and also the town of Amesbury is across said main road less than two miles away.
In the first movie, Prime said they learned Earth's language from the World Wide Web. Some transformers that existed before the Internet was born can speak the English language.
(at around 33 mins) Daytrader brings Starscream's head to the junkyard, and Megatron is later seen holding it when he and his Decepticons arrive. However, the head here is fully intact, when in Transformers: Dark of the Moon (2011), Starscream lost his eye and his head was blown apart by Sam.
(at around 22 mins) Optimus arrives at Cybertron and is distressed, asking "what happened to [Cybertron]?" However the fate of the planet in Transformers: Dark of the Moon (2011) would have been clearly visible to Prime back on Earth.