Installation Verification problems with Mountain Lion and Mavericks

Upfront data:

Built an older Mac Pro to play some old games and run Final Cut Pro HD. Now wanting to update so the browsers work better (even if not perfectly) with today's websites. Running Firefox in order to use the web at all.

Mac Pro 3,1

Dual Quad Core 3.0 GHz Xeons

Currently running Lion 10.7.5

32Gb RAM

Vid Card: Radeon HD5870

Everything runs fine on the current system and OS (Except Safari, which is basically useless on todays website)- even graphic intensive games.

I have tried multiple times (more than 2 dozen) to install Mountain Lion from 3 different download sites, including the download from Apple. All get exactly the same error: "Cannot verify, may be corrupt, delete and redownload." Have tried 10.8, 10.8.2, 10.8.4, 10.8.5 – each multiple times.

I have checked the date, both using Settings and in Terminal. I have tried in Safe Mode. I have reset NVRAM.

No effect.

I then thought: "Well, I'll just skip 10.8 and go to 10.9 Mavericks. Downloaded that from Apple. Same error as with Mountain Lion.

Can't really find anything on searches other than the date issue. And I'm thinking there is no way I have all corrupt files from 3 different sites, downloaded using different machines. Something on my system must be causing this.

Anyone know a resource or have experience working through this?