Failing on apt-get update: The following signatures were invalid

After a little more searching I have found the solution.

Remove/Rename the files associated with the errors:

cd /var/lib/apt/lists/partial/
sudo mv us.archive.ubuntu.com_ubuntu_dists_precise_main_i18n_Index main_COPY
sudo mv us.archive.ubuntu.com_ubuntu_dists_precise_multiverse_i18n_Index multi_COPY
sudo mv us.archive.ubuntu.com_ubuntu_dists_precise_restricted_i18n_Index res_COPY
sudo mv us.archive.ubuntu.com_ubuntu_dists_precise_universe_i18n_Index uni_COPY

Update the signature:

gpg --keyserver keyserver.ubuntu.com --recv 40976EAF437D05B5

Rebuild the software cache:

cd /var/lib/apt
sudo mv lists lists.old
sudo mkdir -p lists/partial
sudo apt-get update

It is probably possible to skip the first step by simply moving the lists, but I figured it best to describe the entire process I used to remove the errors.