Grammar Nerd Question

In the recent post Proofsheet - G&G 04 we wrote "the rest of the roll either exists somewhere else or is lost." We treated "the rest of the roll" as a singular subject and, thus, used singular verbs. Should we have treated "the rest of the roll" as a plural subject and written "the rest of the roll either exist somewhere else or are lost"? Would it matter if we changed the subject to "the rest of the negatives"?

See, kids, grammar is not only important, it's fun!

UPDATE

Craig writes in support and I think I can rest easy now.

"rest" is singular. You are correct. "of ...whatever" is not important to the singularity of the rest.

Whew! Even if the rest of the family members is unsure, we should probably put this to rest. Thanks, Craig!

Dave also chimes in with the rest of the story.

Of course, there are different rules if you are using British English. In America, we would say "Italy has won the World Cup" (or "Italy wins the World Cup!!"), whilst in the UK they would say "Italy have won the World Cup" (or "Italy win the World Cup!!"). We consider a noun that represents multiple items (in this case, the name of a political unit made up of millions of people) to be singular, they regard it as plural.

But of course, they also say "maths" instead of "math," so there's really no reason to take their grammatical rules seriously.

Thanks for the worldview, Dave! I am kind of interested in what the English think of their grammar; the language is called English for a reason, after all. It seems that some people view the resolution of subject/verb agreement for compound nouns as not being so simple. I cannot say that any particular view is clearest, and I do not think the grammar police will arrest anyone for getting it wrong, but in the interest of restoring some manner of prestige to my grammar skills I will continue to wrestle with the issue to find the surest, purest, fairest rule of application. Or, whatever.

OTOH, the use of "maths" seems to be a spelling disagreement more than a subject/verb agreement disagreement. "Maths" is used as a singular, correct?