I work in an at-will state and do not have a contract with a required notice period.
I'm putting in my two weeks notice, but I may only be able to give my current employer half a week in the second week of my notice. Is this a bad thing, or an integrity issue on my part? Will it affect my references?
Edit per comments and updates:
Given that there was no Notice Period specified it is ok to offer a Notice Period, even though you are not bound to give one.
The "default" period is 2 weeks, but if you know you won't be able to fulfill such period I suggest you offer them one that you will be able to achieve (the 1.5 weeks).
Have in mind that you are under an at-will employment, so chances are they might terminate you right there the moment you hand your notice, and thus won't have to serve it.
Original answer:
Is this a bad thing, or an integrity issue on my part?
Yes I fear this may be negative for you.
You contract specifies an exact amount of days you should serve as Notice Period. You should serve those days up to the number specified in your contract, not less.
The company may decide to terminate you right there without having to serve the Notice Period, though, but in case they want you to serve it you are bound to sever the full time.
Since you're not technically obligated to give any notice, this comes down to how it comes across to your employer. That, in turn, will depend on your existing relationship, your position, and why you're only giving a partial week.
In general terms, giving a week and a half rather than two weeks of notice is a bad thing but far more of a misdemeanor rather than a felony. If you have a good relationship with your employer, it is likely that they'll overlook it. On the other hand, if you have a historically rocky relationship where the employer thinks that you're regularly failing to live up to your obligations, this will likely be seen more negatively. If you are in a position where you need to do a large amount of knowledge transfer during your notice period, cutting your notice down will be relatively more negative than if there are dozens of other employees that can perform your tasks already. If you are cutting your notice period down because of some unavoidable situation, that will go over better than if you are giving less than the normal notice because of poor planning on your part.
From the point of view of references, the biggest risk is likely how it affects your eligibility to be re-hired. It is pretty common for a potential employer to ask whether the candidate is eligible to be re-hired at their previous company. It is possible that not giving the normal notice would impact this which would affect references. I wouldn't expect that to be the case but it may be worth having a conversation with your boss if you have any flexibility, i.e.
Boss, I've decided to resign. I want to give a full two week notice but I can only give a week and a half because <>. Is that going to impact my eligibility to be re-hired or your reference for me? If it is, could I mitigate the problem by <>?