This one of those quirks that is highly annoying to HSA owners.
Medicare starts either on the first of your 65th birthday month, or on the 1st of the previous month if your birthday is on the first of the month, or if you sign up within 6 months of turning 65, the start date is set back to the 1st of your birthday month, or once you are beyond 6 months from your 65th birthday, the most backdating is 6 months.
Please see this Medicare webpage to see if that makes more sense.
Now that you know that actual rules, you can decide if you want to rescind your Medicare application (you'll have to look the rules for that up) and start on July 2nd. Now that I think about it, all that matters is that you have HDHP coverage on December 1st; after that, if you start Medicare in the middle of the month, it won't matter to your HSA contribution limit, so long as you had the HDHP coverage all 12 months.
That is, you don't want to invoke the last-month rule (using the full annual HSA contribution limit just because you have coverage on December 1st), because then you have to stay under HDHP coverage through all of the next year, and with Medicare, you obviously won't be doing that.
Does this make sense?
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