4 Way Switching Wiring Diagram Nightmare
First of all, I think someone just did the coloring nonstandard; they simply used red to the fixture for whatever reason. Also, your diagram doesn't show a red between the 4-way and 3-way switches. I'll assume that's an oversight and it's actually there, because if not, you have bigger problems...
That said, there are a couple of things possibly at play. First, the black hot in the big box could be a separate circuit, not intended for the light fixture in question. Outside lights are often on a different circuit. Second, the black line in question could indeed be the power for the fixture, and you have a three-way setup where the power and fixture are in the same box. You'll need to look at the other three-way switch to know:
Three-way switches have two brass screws and a black screw. The brass screws are the travelers and the black screw is the common. In an ideal world, one three-way box would have the power coming in, and the other would have the power out to the fixture. You'd connect the incoming power to the common on one switch, and the fixture's hot wire to the common on the other switch. You then run the red/black traveler in between, connected to the brass screws on both switches, and you're done. The neutrals and ground simply connect to each other all the way across.
In the more-complicated situation of the power and fixture hots being at the same box, the typical method is to use the white of the traveler wire to carry the switched hot:

The incoming power goes to the common screw of the switch in that box
The traveler red and black connect to the brass screws of that switch
The white neutral of the incoming power line connects to the neutral of the fixture
The white of the traveler connects to the hot of the fixture

At the other end:

The red and black again attach to the brass screws
The white attaches to the common (black) screw

So what you need to do is look at the other three-way switch. You should find one of two things: a) a single red-black-white-ground coming in with the white attached to the black screw, or b) two lines, one black-white-ground and the other red-black-white-ground, with the single black wire on the black screw, and the whites attached together.
In the former case, use a continuity tester to check and see if the white coming in with the travelers (the red-black-white-ground from the 4-way switch) gets connected and disconnected by operating the switches. If so, that needs to go to your fixture hot (red), and the hot black line needs to go to the black screw on the 3-way in the big box.
In the latter case, the hot you've found in the big box is unrelated; as I said, likely another circuit. The fixture hot (red) should be on the black screw of the 3-way in the big box, and everything should operate correctly.