Yeah, it'll depend on your jury whether they consider an emulated register to still be a register as described. However, what's super common in these sort of patents is to duplicate the same claim language several times with slight variations to cover all implementation types. If you look at claim 9, it gives a much more broad heading:

"A computer system coupled to receive and respond to computer instructions from a program routine comprising"

In later patents, they got even more clever and just say a "method" rather than a "processor", and explicitly define registers as potentially being emulated in the description (search AVX2 patents if you're curious)

If you don't have a hardware implementation, you are trying to patent an abstract idea, which Alice Corp. v. CLS Bank International found to be invalid.

Nope, an abstract idea has always been invalid. Alice vs CLS Bank found that "on a computer" wasn't a sufficient inventive step to transform an abstract idea into something patentable [1]. This can be used to invalidate a claim, but won't shrink the scope of a claim to hardware only (as then, if software was the only inventive step, it would be a pure abstract idea).

The USPTO certainly seems to think an ISA is patentable, and I haven't seen a court disagree yet.

[1] https://en.wikipedia.org/wiki/Alice_Corp._v._CLS_Bank_Intern...

I'm not entirely sure what you mean. Do you mean if you have a patent that includes hardware, then that patent would prevent emulating the hardware in software? Wouldn't any software patent then be possible, by the simple expedient of describing it in the patent application as running on a custom single-purpose hardware device?

If you come up with a patentable idea, you can specify its implementation in either hardware or software in the patent (or just be ambiguous). However, if your idea alone is unpatentable, you can't add "implement it in software" to make it patentable, according to Alice vs CLS. Basically, Alice vs CLS removes a certain class of software patents, but certainly not all.

Yes, any software patent is possible by describing it as running on a processor. See for example [1], which has the very common claim prefix of "A machine readable storage medium storing a computer program..." Alice doesn't invalidate these patents unless, by removing that text, the remainder of the claim is unpatentable.

Because software patents are still legal, there's no need to attempt to describe them as running on a custom hardware device - you just specify them as software. Specifying custom hardware would unnecessarily reduce the scope of your claim.

[1] https://www.google.com/patents/US6126544