Postdoc that refuses to follow the terms of her contract

Your additional information that you put under the comment section and I think it should be placed in Question also, is that post-doc is paid for by the University, not paid by you/from your grant. So you are not PR, you are an advisor. Nick S, pointed out something that is not specific for mathematics, also physics, material science, applied science, engineering and some types of medicine (biomedicine), I will quote him

a Postdoc position is supposed to be the time when the researcher starts working independently. I don't know how it is in your field, but for me a postdoc wanting to work independently would be a good postdoc.

Dan Fox also pointed out very common thing

A postdoctoral supervisor's primary responsibility is to help the postdoc develop into an independent researcher, if the postdoc is not already an independent researcher. The notion that a postdoc should just take orders and be another cog in a big machine is sad. So is the notion that a postdoc who independently obtains interesting results should add the nomical supervisor as a coauthor.

At my university, it is explicitly written in contract that mentor/advisor needs to be the corresponding author of a paper from a postdoc. I am aware that some contracts don't have this clause, and they can be purely independent. However, if this is the case with her contract, you should stop giving her any resources. If she is not dependable on your resources and still yields some kind of results that are publishable, I am sorry to tell you this, but according to info that you provided in your question, she is in the right.

You need to report progress as it is to head and to above you body. You need to be honest with people that are above you that you don't have any control over her and that you don't know direction of her investigation. Does she know you need to write a report about her progress? I think if she knew that she was evaluated, she would tell you how far in research and what she is doing now. Again you should be more precise about the content of her contract. As far as I know, contracts with universities are very liberal and give a lot of freedom to the postdoc.

It IS a common situation, although there a two parts here: One is work related, she is employed, you are her supervisor, she is not doing her job, you should act in an appropriate way.

The other one is research related, she is producing results (although not together with you), so that might be a criterion to leave her alone. Once again, you shouldn't decide this on your own.

All in all, I would suggest to first talk with the head of department in an informal way, maybe don't even mention her name (unless it is obvious who you are talking about), ask how to proceed. Maybe she produces enough results to allow for a change of contract, giving her more freedom to do her own projects? Maybe she is just lazy and should be fired before getting you into trouble? Either way, you should act before it is too late and you are held responsible for whatever happens or does not happen.

It's unfortunate that you have let it run to 12 months, so that you cannot use the usual probation procedure, which is meant to deal with things like that from an employment perspective. If this were still during the probation period, you could refuse probation (or extend the period) on grounds that she is not fulfilling her contractual requirements.

If the contract is not for a full 4 years, but for 2+2 years, you could terminate the engagement after 2 years and/or suggest this to her.

But after probation, sacking her would

1. require a nasty and involved University-internal process of performance management
2. make the University and you vulnerable to being sued
3. damage your and your University's reputation
4. likely ruin her career for good
5. definitely ruin your friendship

So sacking is not worth it for just 3 more years.

What else could you do? I suggest

1. Keep talking to her regularly about her and your projects. In a research group, this can be naturally arranged by having regular (weekly) meetings of the whole group, where attendence is compulsory and each weak another member reports on their work (or on a conference they have recently attended) etc.
2. Don't consider her as a 'slave worker' (who does what you tell her), but try to support her projects. The best young researchers thrive when they have sufficient freedom to develop their own research agenda and are supported by experienced scientists.
3. Try to have at least one common research project with publications co-authored by both of you (and possibly others).
4. You have one last trump: reference letters. Point out that if you don't know her research well, you cannot really write supporting reference letters. She will know that failing to obtain a good reference from you may well end her academic career.