Will a PhD help me as a startup founder? I'm planning to get double honours in CS and Pure Math. (Link to my previous post.) But I'm very confused about getting a PhD. Will it help me as a startup founder?

The only thing that is stopping me from getting a PhD is time. I have excellent grades(almost perfect GPA). I'm very motivated to launch startup after graduating(maybe after working for a year or so). The only thing that's worrying me is that if I get stuck on a problem while working at my startup and the problem involves some sort of research, Will I be able to do research and solve that problem with my background(CS and Pure Math)? Say that problem is related to machine learning and involves creating a more intelligent machine.
No. 

Disclaimer: I am an academic. I have zero experience with startups. 


A PhD can be a great way to develop cutting edge ideas, but a startup is not the place to conduct research, it's the place to translate proven research into a business. Major conceptual roadblocks should be sorted out and methods proven before you start a company. Consider the difference between Google as a startup and Theranos. Google had a working research artifact that they needed to develop into a commercial platform. Theranos is apparently going to implode because their basic concept has never been proven even after $400 million dollars of funding.
Startups generally aren't founded on an amazing technical idea. A well-motivated undergrad could build the basic technology platforms behind successful startups like Facebook (databases), WhatsApp (sockets programming), or SnapChat (databases plus sockets programming). There are definitely some startups that are successful because they have a great technical edge that no one else has figured out (see again Google for an example), but they're probably the exception. The bigger value of many tech startups is the fact that they serve an unrecognized need. Their future value comes from their mature systems engineering and user base. For example, Twitter serves about 350,000 tweets per minute, which is not a trivial system to create. WhatsApp and SnapChat have a user base that makes them incredibly valuable. 
Research conducted in a PhD program tends to be "basic research". Research conducted for business purposes tends to be "translational research" or "development". They are related but different things. 
A PhD program is probably not going to afford you the opportunity to work on a project of your choosing related to your business. 
If you are able to work on business-related research there will be a minefield of legal questions over the ownership of the intellectual property when you want to monetize your venture. Unless very specific conditions are met and agreements are made before you start your program, it is guaranteed that the university or your adviser will have some claim to the intellectual property that results from your degree. This isn't necessarily a deal breaker, but it's probably beyond you if you don't currently have the resources to hire a lawyer. And you're not going to get the university to sign away its rights. 
There are huge opportunity costs to getting a PhD. You will only have time for a business alongside a PhD if you are extremely dedicated. It is not uncommon for PhD students to work 50-60 hour weeks. If you work on federal research your stipend will be around $25,000-$30,000 per year, which you can easily double as an undergrad doing low-level CS stuff at a company. If you are a hard worker in the right place you could easily triple that number after the five or six years it would take you to get your degree.