I found the Twitch Launcher logs complaining about the HTTP 403 forbidden status error, however, the Minecraft Launcher acted differently when trying to recreate the logs and didn't even attempt to download the libraries. Previously, it ran into the same HTTP 403 status error on the same file.
Twitch Log: https://pastebin.com/p1RJZ9UW
Minecraft Log: https://pastebin.com/hs7wQR7k
PS: I've seen SSL failures before using the Minecraft Launcher's built-in Java runtime to manually install forge instead of a system Java installation. I'll post a forge install log if I see this again.
Edit 1: Generated the forge install log. https://pastebin.com/j09sMuL6
(Built-in Java version is 1.8.0u51)
Edit 2: The built-in Java runtime seems to be an unrelated issue. I stealth upgraded it to 1.8.0u211 (current version at this time) by replacing both bin and lib folders and both launchers continue to fail the installation. The Twitch Launcher requires multiple attempts to make it think it installed Forge and allow me to run the Minecraft Launcher to the point where I got the HTTP 403 error previously. Here is the resulting log. https://pastebin.com/yEBEa7Gi