Is there a doctrine or Biblical justification that explains why God chose the number 7 to be so significant? Do any major traditions in Christianity hold any sort of formulated doctrine or is there some Biblical justification tossed around that explains God's a priori reason for choosing 7 to be significant was? 

I'm specifically curious about the source of 7's significance, not the how (in other words, I'm not looking for examples of significant 7s (covered in Why is the number seven special? and on Judaism.SE here) or for how these play out, but for why God picked the number in the first place.)

As I understand it, the significance of the numbers 1 and 3 can both be derived from the Trinity. One God, three Persons. But what is the source of 7? Why God chose seven instead of, say, nine. Or five or eleven, which are also primes (in the mathematical system we use). God seems to have chosen 7 to be significant before Creation. He chose 7 to be the number of days in a week. This means that any significant 7 post-creation (including the length of a week) is a significant 7 for the same reason. Before God created the universe, He picked out 7 to be one of His significant numbers.

To clarify a bit further, if God decided that 9 was the number of perfection for whatever reason, then we probably would have had an 8-day work week and the 9th day would be the Sabbath. The number of the beast would probably have been 888 instead of 666. Pentecost probably would have come 82 days after Passover (50 = 7*7+1, 82=9*9+1).

If there is truly no source for this and no Christian tradition has forwarded any explanation, the answer may very well be "none". However if more than one doctrine exists, please explain what the different suggestions are and how they end up on different tracks.
I don't think you'll find dogma from any major tradition insisting that God pre-ordained the number 7, that it's integral to His nature, or anything of the sort. In fact, scripture suggests the contrary.  Consider first God's consecration of 7 as the symbol of perfection. It's the final event (or detail) of the first creation account.

Genesis 2:1-3


  1 Thus the heavens and the earth and all their array were
  completed. 2 On the seventh day God completed the work he
  had been doing; he rested on the seventh day from all the work he had
  undertaken. 3 God blessed the seventh day and made it holy,
  because on it he rested from all the work he had done in creation.


So, right off the bat, regardless of whether we assume the creation story is allegorical or literal, we can already start to make a case that God consecrated the number post-creation. He works for six days, and once everything is to His liking, he declares perfection, which just to happens to be on the 7th day. Seven thereby symbolizes perfection. Nowhere else (that I can think of) does God directly reveal 7 as a symbol of perfection; it is seemingly deduced from creation, God's reflection on the 7th day that His work is perfect.

Those preferring the allegorical reading might suggest that the story takes place in 7 days because the Jews already used 7 to symbolize perfection, or because certain prevalent natural harmonies in the universe made 7 significant, etc.. Those preferring a literal reading can start to sense that God didn't pre-consecrate 7; that consecration occurs distinctly after God creates everything. And in either case, the consecration of 7 as perfection occurs at a particular point in time in the story -- in fact, it's the point in time itself, according to the story, that defines the number of perfection, which "ends up" being 7.

Now, one could (and should) object here and suggest that, since God is omniscient and omnipotent, He could simply have planned to make all things in 6 days and declare perfection on the 7th. He was eternally aware of 7's perfection. And that's a fair point.

Thus, we turn to Jesus' own words to weigh in on our speculation. The first story in gospel of Mark wherein Jesus is challenged with a question about the Sabbath is significant. Upon being caught disrespecting Sabbath law, Jesus rebuts with a pretty revealing statement.

Mark 2:27-28


  27 Then he said to them, “The sabbath was made for man,
  not man for the sabbath. 28 That is why the Son of Man is
  lord even of the sabbath.”


This statement points all the way back to creation. The Jews revere the number 7 as the symbol of perfection on account of God's mandate for a Sabbath. And now Jesus is telling the Jews that God made the Sabbath for them.

The point is simultaneously subtle and glaring. God doesn't need a Sabbath. And God doesn't need 7 to mean anything. We do. Therefore, for our sake, God chooses the number most beneficial and recognizable to our nature to signify perfection and denote cycles of rest and worship.

God choose seven to serve human nature, not His own. More to it, God doesn't need symbols or special numbers. We do. God just gives us what we need (or ask for) to help us stay on track.