Time Paradox Incident

The Time Paradox Incident was the Time Lord designation for the event in which the Third Doctor prevented the assassination of Sir Reginald Styles. (TV: Day of the Daleks, PROSE: Dalek Combat Training Manual)

Led by the Gold Dalek, the Daleks used time travel to invade and occupy Earth again, their previous attempts having met with failure, exploiting the outbreak of World War III on Earth in the 20th century. Guerilla fighters opposing the Daleks' occupation in the 22nd century used time travel of their own to attempt to avert the war that had given the Daleks their opportunity, unaware their actions would actually precipitate the conflict and paradoxically enable the very invasion they were fighting. Their visits to the past embroiled UNIT and the Third Doctor currently serving as their scientific adviser. (TV: Day of the Daleks)

Prelude
The Time Lords understood that the Daleks enacted the Time Paradox Incident shortly after discovering a means to make short hops through time. (PROSE: Dalek Combat Training Manual) Human historians from the far future suggested that the Daleks were motivated by their belief that, by invading Earth during weak or tumultuous points in its timeline, they could disrupt its future. If they instigated catastrophic change, it would stunt the expansion of humanity and exterminate the human threat forever. With Earth's future history rewritten, the Daleks could avert their protracted war with the human colonies of the First Great and Bountiful Human Empire, essentially allowing the Daleks to win before the conflict ever began. (PROSE: Dalek: The Astounding Untold History of the Greatest Enemies of the Universe)

By one account, the Daleks had developed the Time Vortex Magnetron which they would use in the Time Paradox Incident by the conclusion of the Mechon Wars which had been fought in the aftermath of the Mechonoid Incident. The "reinvasion" of Earth was sanctioned by the Dalek Prime himself. (PROSE: The History of the Daleks)

The Gold Dalek, a Supreme Dalek and member of the Dalek Supreme Council, (PROSE: Dalek: The Astounding Untold History of the Greatest Enemies of the Universe) allied with the Renegade Time Lord known as to facilitate Operation Divide and Conquer, (TV: Frontier in Space) a plan approved by the Dalek Emperor himself. (AUDIO: The Dalek Conquests) The Daleks were thus granted access to the Master's knowledge of temporal mechanics. Further research was carried out by the Gold Dalek, (PROSE: Dalek: The Astounding Untold History of the Greatest Enemies of the Universe) resulting in them "discover[ing] the secret of time travel" and learning how to change history on a larger scale than was previously possible. (TV: Day of the Daleks)

However, the Gold Dalek's decision to ally with the Master having led the Daleks to a defeat on Spiridon that set back the Empire's war efforts significantly, even causing them the loss of another Dalek Supreme. (TV: Planet of the Daleks) As the Gold Dalek felt responsible for the failure, to redeem itself, it orchestrated a time travel-based mission to conquer Earth in the 22nd century, creating an alternate timeline. (PROSE: Dalek: The Astounding Untold History of the Greatest Enemies of the Universe) By other accounts, the alliance with the Master came after the time travel-enabled invasion of Earth; the Gold Dalek did not recognise the Third Doctor during the invasion but did during the alliance with the Master (TV: Day of the Daleks, Frontier in Space) The Time Lords also stated that the invasion of Earth came before the Dalek alliance with the Master within the timeline of the Daleks. (PROSE: Dalek Combat Training Manual)

Legacy
Informed by the failure of the Gold Dalek, one of the remaining Dalek Supremes later organised a smaller-scale time mission, (PROSE: Dalek: The Astounding Untold History of the Greatest Enemies of the Universe) going further back along the timeline of the Doctor and attempting to execute him in his first incarnation. (TV: The Chase) Other accounts suggested this hunt for the First Doctor occurred earlier in Dalek history. (PROSE: The Chase, Mission to the Unknown, et. al)

The Time Paradox Incident made the Daleks more cautious about attempting to interfere with time, while alerting them to the fact that such alteration was theoretically possible. From this, the Dalek Prime realised that the Daleks could not go back and prevent the historically recorded destruction of Skaro caused when Davros activated the Hand of Omega, and so arranged for the planet Antalin to take the place of Skaro as a decoy. (PROSE: War of the Daleks)

By one account, it was following this incident that the Daleks were embroiled in the war against the Movellans. (PROSE: The History of the Daleks)

Though the alternate timeline in which the Daleks conquered Earth was averted by the Doctor, The Dalek Conquests suggested that the incident had brought the Daleks to the attention to the Time Lords, who would later assist the Third Doctor in the Spiridon campaign. It was further suggested that the incident motivated the Time Lords to plan their "future strategy", leading to the Last Great Time War. (AUDIO: The Dalek Conquests) Indeed, the Time Lords saw this incident as the point in which the Daleks "master[ed]" time travel. (PROSE: Dalek Combat Training Manual) It was during the Time War that the Daleks, through the Overseer, rewrote history so that they used Ogron mercenaries during a number of skirmishes with the Doctor, but did not rewrite it enough so that their outcomes changed. As a result, the Eighth Doctor recalled the Ogrons' presence in these pre-war incidents. (AUDIO: Planet of the Ogrons)

The Time Paradox Incident was among the Doctor's encounters with the Daleks which were tactically assessed by the Time Lords. Recalling that the humans constructed their own time machine from stolen Dalek plans, the Time Lords deduced that the Dalek time technology was based on their own, presumably as a result of them having captured, or having had ample time to study, a Gallifreyan time capsule, much to their concern.

Their technical division postulated that it may be possible to retroactively insert a "Trojan Horse" program into dematerialisation circuits at a time period prior to the Daleks' acquisition of the technology, thus giving the Time Lords a shut-down option that would deprive the Daleks of time travel capability. Whilst the War Council agreed that it would be an effective defense, the possibility was recognised that the Daleks could discover such a program and potentially use it against the Time Lords. A subcommittee looked into the ramifications of this course of action and was to report back once its findings had been analysed. (PROSE: Dalek Combat Training Manual)

In the post-Time War universe, this incident was covered as a part of known Dalek history in The Dalek Conquests, a documentary which was itself produced following the Van Statten Incident on Earth in 2012. (AUDIO: The Dalek Conquests)

Alternate universes
In the Unbound Universe, where the exiled Doctor arrived on Earth in 1997, the Brigadier had seen three Daleks kill 47 of his men in 1972, which was later cited as evidence of his incompetence. Since then, the Daleks had visited Earth "several times". He recalled the initial encounter when he saw the Daleks on Skaro. (AUDIO: Masters of War)

Behind the scenes

 * Episode four of Day of the Daleks was originally intended to include a confrontation between the Doctor and the Daleks, in which the Daleks explained how they destroyed those of their number who were infused with the Human factor in The Evil of the Daleks, and turned their attention to conquering Earth by means of time travel. But this scene had to be edited out, due to the episode overrunning. Bringer of Darkness would later establish the Doctor having been made aware earlier, whilst still in his second incarnation, that the Daleks had survived "the Final End".
 * The Discontinuity Guide claimed that the Daleks, following the defeat of the 22nd century Dalek invasion, completed their time travel program, which they had began with Theodore Maxtible, and waged a successful invasion of Earth in the late 21st century, whilst they themselves came from no later than the 23rd century, long before the Draconian Gambit. When their work was undone by the Third Doctor, the Daleks pursued the First Doctor in an attempt to neutralise his "future intervention". It is further noted that, since the Fourth Doctor inadvertently changed Dalek history so that Davros survived, this event would have occurred vastly differently if it happened at all in the new timeline.
 * The Dalek Handbook suggested that the Time Lords had ignored the Daleks' original wave of conquests and their development of time travel technology before the combination of the Daleks' large-scale efforts to alter history in the Time Paradox Incident along with the Master's alliance with them and the Doctor's subsequent appeal for help motivated the Time Lords to notice the Daleks, foreseeing the potential future which led them to enacting the Genesis Incident.
 * AHistory dates the incident, from the Daleks' perspective, to roughly 2172, following the launch of the first DARDIS which itself followed the failed invasion.