A time paradox or temporal paradox is a theoretical 'conflict' between past and future events that occurs when mutually-exclusive changes are made in the timeline. Since the games in the Metal Gear series do not take place in chronological order, the player can create a paradox in the Metal Gear timeline by causing the death of a significant character. The Time Paradox screen, featured in Metal Gear Solid 3: Snake Eater, Metal Gear Solid V: Ground Zeroes, and Metal Gear Solid V: The Phantom Pain, either follows or takes the place of the traditional Game Over screen, respectively.
Metal Gear Solid 3: Snake Eater
If Naked Snake is killed, the game comes up with the traditional Game Over page displaying "Snake Is Dead." After a little while (or quickly if the player selects Exit instead of Continue) "Snake Is Dead" changes to "Time Paradox." This is because Naked Snake is revealed to be Big Boss at the end of the game, the genetic father of Solid Snake, Liquid Snake and Solidus Snake, and if he were to die, all subsequent events could not take place. If the Game Over screen is triggered via the use of a fake death pill, the point at which the "Time Paradox" message is completed marks the point at which the fake death becomes real. In this scenario, the player can access the gadgets menu and select the revival pill to quickly reverse this process so that Snake can stay alive.
A grenade can be thrown through the window of the room Sokolov is being held in, bringing up the Game Over page displaying "Mission Failed." His death compromises the Virtuous Mission, as he would not be able to complete the Shagohod.
Ocelot can be killed if Snake shoots, stabs, or throws a grenade at his unconscious body near the end of the Virtuous Mission or kills him while he is on the catwalk during the fight with Colonel Volgin, bringing up the Game Over screen and the message "Ocelot Is Dead." As with Snake, Ocelot causes a Time Paradox because of his involvement in future events (such as being the one responsible for the loss of Naked Snake's right eye). Ocelot is also one of the founding members of the Patriots and is the right hand man of Liquid Snake and then Solidus Snake. His death would also impede a future Old Snake from trying to kill Liquid Ocelot in 2014 because he is already "dead."
This is hinted at by the achievement/trophy in the HD version of Metal Gear Solid 3, titled "Problem Solved, Series Over", obtained by causing an "Ocelot is Dead" Time Paradox. After all, knowing Ocelot's actions in the future would make killing Ocelot in Metal Gear Solid 3 a logical step. But in doing so, would make the series stop at Metal Gear Solid 3, canonically the earliest point in the timeline.
If EVA is killed, a Time Paradox is created, bringing up the message "EVA Is Dead." She is essential for the completion of Operation Snake Eater, as she is the one who pilots the WIG out of Tselinoyarsk, as well as fighting the Shagohod and escaping Volgin's forces. EVA also plays an important role in later events, becoming a founding member of the Patriots and the surrogate mother to Solid Snake and Liquid Snake, and serves a part in the Guns of the Patriots Incident under the codename Big Mama.
Metal Gear Solid V: Ground Zeroes
Should the player be killed during the "Ground Zeroes" mission, small text under the Militaires Sans Frontières logo reads "Time Paradox." It should be noted that this only happens during the main mission, due to the side ops in Ground Zeroes being labelled "pseudo-historical recreations."
Metal Gear Solid V: The Phantom Pain
During the opening of the game, Snake (Ahab) and Ishmael escape the hospital during an assault. If the player dies, the text reads "Mission Failed", whereas if Ishmael dies, the text reads "Time Paradox." This acted as a subtle hint as to Ishmael's true identity.
In Mission 2 "The Phantom Limbs", if player takes longer than three days to reach Kazuhira Miller, a different cutscene is triggered with Snake discovering Miller dead, and the mission fails with the "Time Paradox" text.
In Mission 12 "Hellbound", if Huey Emmerich is killed, the mission will fail with the "Time Paradox" text.
In Mission 23 "The White Mamba", if Eli is killed the text reads "Time Paradox" instead of "Mission Failed." This is a hint that he is a young Liquid Snake.
In Mission 29 "Metallic Archaea", if the helicopter with Ocelot in it is destroyed, the mission will fail with the "Time Paradox" text.
Even though Venom Snake plays an important role in the Outer Heaven Uprising, the "Time Paradox" text does not appear if he is killed.
- Metal Gear Solid 4 Database ("Time Paradox")