Why did my soy (Silk) milk suddenly become as viscous as rubber cement?
Soy milk can spoil, just like ordinary cow's milk. That seems to be what happened.

While spoilage in cow's milk is usually souring, and the smell is unmistakable at fifty paces, soy milk spoils by turning gooey.
I'm not sure about the details, but it's something in how the proteins react to oxygen. In soy milk, if I'm not mistaken, they turn into longer, stickier chains.

As for why it spoiled, it was probably just open for too long. Even in the fridge, soy milk has a life of about 4-6 days once opened. The seal may have been damaged somehow even before you actually opened it, allowing air into the carton to cause the spoilage.