Theory:SJA television discontinuity and plot holes/The Empty Planet

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  • Rani acts as though Gavin is so much younger than she is. He can't be that much younger than her?
Rani and Clyde are around 17. Gavin states he is 13 in Part 2. It would be perfectly normal for her to act the way she does.
Actually, 13 year olds and 17 year olds socialise perfectly normal with each other.
It varies. Some 13 year olds and 17 year olds might socialize normally, many don't.
The Robots left him there legally. The Doctor does not adhere to legal rules, especially when it affects the course of history itself.
There is nothing physically preventing him. The law says he can't leave Earth - that doesn't mean he actually can't. It just means he will be punished if he is caught doing so. In this story, the Robots just chose to leave him on Earth to avoid getting into trouble. The Doctor on the other hand, ignored it.
The Doctor also simply may not have been aware of the edict, presuming Sarah Jane had not given him detailed synopses of her adventures.
  • Rani and Clyde say they have no 'robot dog'. They are listing who has disappeared and correctly say that Mr Smith has gone as he is in fact alive so would also have been removed, but K9 is not alive; so there is no reason to assume that K9 would have vanished. The just assume he's gone which is stupid especially as they hadn't bothered to check. It would have been difficult to check him anyway as they would have to go to the university themselves, but they can't just presume he's not there without checking.
As you mentioned in your above "statement", they simply say that they have no robot dog. They don't. Whether or not K9 is still on Earth, he is not there to help them.
And even if he were, the story establishes that electronics (at least initially) were disabled. If Mr. Smith was disabled, then likely so was K-9. And K-9, being self-sufficient, would certainly have found some way to contact Rani or Clyde if it were possible.
  • Why does Mr Smith not work once everyone has been removed from Earth?
Mr Smith is actually a sapient crystal called a Xylok encased in the computer interface. The Automatons removed every sapient organism on Earth as legally possible via the warp-shunt, which included Mr Smith.
Or he was disabled along with the cell phones and computers.
  • Gavin denies the existence of aliens and says he's never seen one. Yet he would have lived through the events depicted in TV: The Stolen Earth/Journey's End along with everyone else. In a similar vein, the sudden loss of 90 minutes shouldn't have confused Haresh so much given that he too would have lived through the events relating to the Dalek relocation of earth.
The Time Field apparently erased those events from history, according to the story arc of TV Series 5. Note that SJA: The Vault of Secrets also shows that belief in aliens is no longer widespread as of 2010.
  • Rani and Clyde have traveled in the TARDIS as recently as the previous storyline. Yet they have not acquired the ability of the Doctor's companions to understand any language.
According to TV: The Christmas Invasion, the translation ability is a function of the TARDIS's Telepathic circuits and requires the Doctor's participation to an extent. Since the Doctor and the TARDIS are not on Earth at the time, this is likely why Rani and Clyde do not understand the alien language.