Mailing the story:

Thanks, Johnny Hughes. He commented on my previous CentOS odyssey, which led me to install version 3.9 on my VIA C3-equipped test box. He said CentOS 4.5 would run on it. I had previously tried the 4.4 live CD, and that wouldn't boot, so I never bothered to burn a 4.5 install disc. Johnny was right. I did burn the first CentOS 4.5 disc, and typing i586 at the boot prompt worked.