How can clients be sure I'm using the Open source code?

GOB

I've been hosting a website on a server. If I publish the website's source code (i.e /var/www/my-website/*) in an open-source manner, how can users be sure my website is running off of that code? If I had the website's source code on github, for example, couldn't I just use different code on the actual server? Would PGP signatures help mitigate this? Or PGP-signed file checksums?