How does the Xbox figure out what time to stamp achievements?

As long as you're signed into Live while playing it's connected to the internet and can pull data about what time it is.

Thanks for the quick reply!

All I can tell you is that "Everyone" that I know or have talked to that has this skin has the achievement unlocked AND has a time stamp on it. Everyone that does not have it unlocked either does not have the achievement or has it and DOES NOT have a time
stamp on it.... This is the way it is as of right now and yes I have the skin and the achievement WITH A TIME STAMP

my achievements for dismantle 50 items, decode 25 items and a change of heart took 9 days to unlock. i should have had those achievements earlier then the time stamp says on my GS card. this seems to be a common thing with xbox one, achievements taking days
to unlock. hopefully MS figures out why this happens and fixes it.

Encharted Arms has no online only achievements.

It's nothing to do with me,i don't work for Microsoft.

You don't seem to understand,it is impossible to have online only achievements without a time/date stamp unless you mod that achievement.All of the excuses you describe can't happen.

If you are connected to live you will always have a time/date stamp next to the achievement you unlocked.You have one of the vidmaster achievements on Halo 3 without a time/date stamp which is a dead giveaway,that achievement can't be earnt offline,you
have several "win xxx amount of ranked online games" achievements without a time/date stamp on a Basketball game,the only way to unlock those achievements without a time/date stamp is by modding

The evidence is there for everyone to see,your Gamerscore has been modded despite what you say & you will never get it back.

Microsoft haven't "screwed" you out of your achievements,the person tha modded them did that.