What to do if a co-author is delaying announcement of results? I am finishing writing a joint paper with a collaborator and we are both extremely excited about the results. However, we cannot agree on when to announce the results. 

In my opinion the paper is essentially ready and we should announce the results very soon. The field in which we are working in is booming right now, there are several brilliant, quick people we are competing with. We have a breakthrough result and in my field it is not unusual to see people arrive at the same results simultaneously and independently. Seeing a lot of activity and knowing more or less what people are working on I feel that we need to secure priority essentially immediately. This would mean e.g. circulating the current version of the preprint among colleagues or, ideally, posting it on a preprint server.

My co-author on the other hand would like to explore other potential applications of our results. The main result of the paper is complete, and the only question is how many applications we can have. We currently have 5 very nice applications but my co-author would like to look for more. If they are found they would be certainly good, but at this point it is more of a fishing expedition. However, because of this my co-author demands that we announce our results three months from now and is insisting on a gag order until then. (In our field it is not uncommon to lose results to dishonest researchers who learn about your result early and can try to publish/attach their name to it before you).

Basically, I feel like we are sitting at a roulette table with a million dollars but instead of cashing in we are waiting to see if we can win another 100k.
I would really appreciate some advice on how to reasonably resolve this situation.
I think you should push harder to get the paper onto arXiv. The important thing is to have a rational discussion with your co-author where you discuss your worries and try to address them. Here are some additional arguments you can try to use to convince your co-author:


If someone discovered your main results and puts it on arXiv right now, you lose potentially a lot. (Though not everything; if the result is reasonably deep posting your paper one or two days after the other group will result in recognition as "simultaneous discovery" at least.) 
If you posted your paper and someone else derived good applications from it, you still gain some increases in your h-index (probably). 
Posting things on arXiv does not mean the paper is immutable. You can continue to think about further applications and revise the paper as necessary. It is only after you submitted the paper at a journal for review that you really have to start worrying about making big changes to the manuscript. 


But absent you finding out exactly why it is that your co-author is reluctant to post the manuscript on arXiv, the brainstorming sessions with strangers on the internet can only go so far. 

I probably am not familiar with your subfield of mathematics, but this mentality of "In mathematics being able to apply your result is what makes it more sellable, papers are sometimes turned down if the result itself is too technical but there are no applications in sight" is foreign to me. In PDEs frequently a technique is developed to solve one or two problems initially. And the originators then work very hard the next few years to extend the technique to apply to many different problems. It is surprising that you are still worrying about the number of applications when you already have five!