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Unfortunately, I have to say I noticed a few holes in the plot myself, even on first viewing. The ones already mentioned, though I'm sure they can be handwaved away for the most part. However there are a couple more serious ones (to me) that I saw:
And they have to do with the Zygons. When Elizabeth I meets her Zygon doppelganger, it's clear she has never encountered the creatures in her life. Shortly afterwards she reappears to capture the Doctor(s) and remarks that her alter ego is 'indisposed', giving the impression that she herself is the Zygon commander enough to fool the Doctor - who is not only familiar with the aliens but the Tennant version is, ah, familiar with the real Elizabeth (granted, he has trouble telling human from Zygon apart, but still). She even appears comfortably familiar with their plans and technology. Yet in a sudden switch that takes everyone present by surprise, she reveals herself to be the real Queen Elizabeth, having killed the commander earlier. Did she really learn enough about the Zygons, their technology, their plans etc, in the time between running off and reappearing in the forest?
Come to that, if she was the real Elizabeth all along, why did she keep up the ruse of being the Zygon commander when the Doctors are released from the cellar? As far as she knows at that point, the only people present are the Doctors and Clara. Okay, she might have been suspicious of Clara, but the Doctors didn't seem to mind her being around and there could have been the usual introductions etc.
Then we come to Kate Lethbridge Stewart, which is sort of the reverse of the above. It turns out that from the very start, from the moment we meet her, she is a Zygon duplicate, revealed only when she and Clara enter the Black Archive and are met by more duplicates. Was there really a need to unmask herself at all at that point, if the objective was to retrieve and reactivate the Vortex Manipulator? Why even co-operate with the Doctor's plan at all; he was trapped in the sixteenth century, safely out of harm's way for their invasion?
Further to that point: how did she connect the Doctors being taken to the Tower, the Vortex Manipulator and a string of numbers etched into the cellar (i.e., how did she know what to look for and where, plus their significance)?
Those are the main ones that leaped out at me, at least, striking me as jarring and perhaps worth another draft to iron them out. On the other hand, there were many points that I picked up on as touchingly clever:
The business at the end about the Doctor wistfully musing about being the curator himself is actually foreshadowed earlier in the episode. Elizabeth's letter to him reads, in part, "In this capacity I have appointed you as curator of the Undergallery". So he already is the curator and could very well become Tom Baker as they discuss.
Tom Baker's cameo as the curator was delightful, but did anyone notice his earlier input to the episode? As the Doctor, Clara and Kate enter the painting room for the first time, about 25 minutes in, you can just about make out Tom's voice finishing the line about something being "untouched".
I want to see more John Hurt! A spin-off Time War series perhaps? He was born to play the Doctor! Continually berating his successors for the more annoying of their mannerisms - delicious!
Billie Piper was great in this. I always found Rose annoying and practically cheered when she left, but I really enjoyed this Bad Wolf version.
I think they missed a trick at the end, actually, although the dream bit with all the Doctors was a nice touch. If you've seen An Adventure In Space And Time, there's a moment at the end when William Hartnell is standing at the console and looks up to see Matt Smith looking back at him. It's a nice little moment, reassuring Hartnell that his Doctor was in safe(ish) hands. I think it would have been a nice little touch if there'd been a mirror of that moment in this story, with Smith at his console looking up to see Hartnell looking at him. That way the two specials would connect on that level. Ah well.
At the age of five, Skagra decided emphatically that God did not exist. This revelation tends to make most people in the universe who have it react in one of two ways - with relief or with despair. Only Skagra responded to it by thinking, 'Wait a second. That means there's a situation vacant.'