What does key-maker's depiction of time window as a multiple of pi imply  about the matrix architecture?

Yes, 314 seconds is a clue for the number Pi. There are a few reasons: 1. Pi is synonymous for a circle, i.e. a closed loop where the end equals the beginning. The door that Neo must enter both destroys Zion and ends the current version of the Matrix, but also starts them both anew.  So Pi is a little clue, possibly a danger sign, that the door takes us back to the beginning. Every past Chosen One has decided to save humanity but also to enslave them. The Path of the One is a circular trap that no one can escape from. 2. Pi contains the code for everything that is, ever was, and ever will be.  The door leads to the Source.  The Architect is not the Source, he is just the antechamber, the room before the main event. The Source lies beyond, behind the door that Neo did not chose. As it's name suggests, the Source is that from which all Machine functions originate. Pi is a normal number, meaning that its decimals carry on infinitely but never repeat. Somewhere in Pi is every possible combination of numbers. As computers convert everything to numbers - photos, music, video, etc, then Pi contains every file of data ever.  Vi Hart explains in "Are Shakespeare's Plays Encoded in Pi?" 3. It's used to sell video games.  That's right.  This entire sequence of Neo rushing through the door while Trinity and Niobe risk their lives to destroy power plants, everything is a tie-in to the (horrible) video game Enter the Matrix.  So if it feels to you like this little subplot is empty and unexplained, now you know why. Most of the story is contained in Enter the Matrix.