Upon completion of the Dog Show meta you receive this runaround. It tells you to start in Lobby 7, so do so.
Walking east out of Lobby 7 takes you into the Infinite Corridor. Past posters of upcoming dog shows? Well, posters of something.
"When you get to a pair of Doric dogs..." - Hopefully realization dawns at this point. The runaround has gone so much to the dogs that they have taken over its text and replaced many of the words, a la "smurf". Anyway, the columns you pass on your way out of Lobby 7 are Ionic. The Doric ones are at the entrance to building 11. But you don't pass through them. The runaround tells you to turn right, which goes into building 3.
In this corridor there is a display of books and a black skull in the floor and two wooden benches. When you reach the end (really the only right turn available unless you go into a room), turn right (into building 1).
The thing you pass on the left with all the dogs is the sign just before the intersection, identifying the department; each dog is a word or number. You can turn either left (into more of building 1) or right (into building 5) and pass through a doorway either way, but only going into building 5 takes you under a large prime dog (the 5). The dogs you pass on both sides are ships (and if you did this runaround on Sunday, the chained dogs were actually unchained, but the chain was still there and I hope you could figure out what was meant). The only thing you can emerge again under is the dome, and when you get to it, a sign across Lobby 7 tells you that you are there.
The shiny dog that sometimes talks is the "Under the Dome" kiosk. You can go behind it either through the double doors to the right, or by entering the building 7 corridor left of the kiosk and entering the first door on the right, in either case ending up in a stairwell. If you head up, you will not pass any red dogs, so head down. Don't go out the doors in front of you, but turn around and pass an array of red fire-control boxes, through the doors on that side, and in front of a long five-word sign.
You can't turn right here - the office the sign is for is locked - so turn left, and then left again, but not at the immediate left, which dead-ends quickly. At the next left is a sign informing you the Infinite Corridor is there. You'll briefly pass outside (where the road passes through) and see blinking lights before you reenter the corridor. After the swinging doors you'll pass three vending machines, two grey waist-high bins, and four mail boxes, all on your right.
The "dog up" are stairs up, which are on your right, but you need to take the next left into building 10, where you pass a vending machine of a different brand, and two posters, a rotated one and an upside-down one.
The only ramp goes down into building 13, full of all sorts of machinery! At the end of the corridor, there is an area on the right that is obviously caged off which you shouldn't enter if, as sometimes happens, the door is left open. Instead turn left and then very soon right, passing the ladder to go up the stairs, out the doors into a vestibule, where one door has a red-on-white sign reading EXIT HERE.
Do so to reach the outdoor corridor which, due to construction, is walled off so you can only turn left or right. At the left end of this corridor you can see an eight-sided STOP sign, while the right end features a four-sided construction sign. Turn right. No matter how short or long your steps are, the only way you are likely to be able to turn right again after twenty-dog steps is to enter Lobby 13.
The last part is just there to make you stop and wait. A man wearing a bandage wrapped around his head approaches you from the other side of the lobby. After you identify yourself to him, he hands you a TOP DOG medal and a slip of paper with a link to a weird video (where the man identifies himself as Lucky and tells you a little about what is going on). Your team also receives an email at this point with a link to the video and also a link to the next round. (If you did this after the video and round were released to all teams at 8 PM Friday, you only got the medal.)