I suspect that Google prefers to remain ambiguous on how this works in case that information can be abused (although I cant figure out how). 

Perhaps the truth will slip out some time, probably by accident.

Thanks to everyone who contributed here.

You also got some great feedback from some the the best SEO studs in the business IMHO.    
+Enrico Altavilla did an awesome, near overboard, but certainly very insightful job of explaining PR distribution then and now.

+Alistair Lattimore ,too, has more SEO knowledge in is left pinky than I have in entire body….and many others that I follow and trust.

Bottom line I see here is that you aren't going to get that answer you want from big "G".   So what do you plan to do now?

You did get a pretty definitive consensus from a great group of superb international SEO's opinions.    Opinions that have been gathered from their own trials and tribulations,    testings and such.    Just curious if "G" won't give you your definitive answer you crave, what you next step will be? 

[FWIW I agree with you wholeheartedly however]

Also, if a website owner needs an SEO justification to fix a serious usability issue, then in my opinion they are not doing a good job at making the site usable and at optimizing it for search engines. Their approach to SEO could hugely improve if they'll start to observe things from the point of view of the user.

I need to provide an SEO justification for cleaning up broken internal links on a site, and TBH I cant find anything I can refer to.

Even Google, in the past, complied with the real meaning of that directive and when they unilaterally changed the way the directive was interpreted, they kept its original name in their documentation (probably because the NOINDEX directive is a well known name that the webmasters were already accustomed to).

This change was a bit confusing to webmasters, but it's not entirely Google's fault. Technology has changed a lot since the first web search engines were created, and those old directives were named after the internal structures used at the time. There was an inverted index (a list of associations between words and documents) and an archive (a repository of the raw contents of a resource), so the instructions "NOINDEX" and "NOARCHIVE" made sense. Maybe, since then, the Google technology has changed so much that an inverted index and an archive do not exist anymore and it's no more possible to follow the original meaning of those directives…

…or maybe the internal Google definition of "index" is still "a list of associations between words and documents" and the change of interpretation of the NOINDEX directive was just a convenient way to keep more document in the inverted index. The following video by Matt Cutts seems to confirm that when they talk about "the index" they are still referring to the industry-wide common meaning, an inverted index:
https://www.youtube.com/watch?v=KyCYyoGusqs

So, every time I read about "the index" I assume that they are talking about an inverted index.

My reply to Gary was about a different aspect, though: I was saying that that anything index-related (the index itself, the indexing process and the "differently interpreted" NOINDEX directive) has nothing to do with how the PageRank flows. So, it's not correct to state that the PageRank will not flow because a resource is not indexed. It doesn't flow because a resource is not in the link graph.

The use of NO_INDEX_ as the value to indicate to search engines that they should stop a page from being returned in search results always seemed strange to me, something like NODISPLAY might have been clearer.

While the answer seems obvious to you, it is not so obvious to people who, just a few years ago, observed Google re-distributing the non-assigned PageRank between the do-follow links in a page.

If the distributing algorithm is:

A:
1) Ignore all the links that point to 4xx resources
2) Divide the page PageRank between the (remaining) links

…you get a very different result from:

B:
1) Divide the page PageRank between the links
2) Don't assign any PageRank to the 404 resources

We know that the correct way to do it is "B" but you should't pretend that there is only one way of managing the distribution of PageRank and, more importantly, since Google has changed their minds about how to handle the follow/dofollow links, asking the same question about 404 resources is perfectly justified.

So, yeah, it's "B", but don't forget that in the past Google temporarily applied the "A" approach to nofollowed links.

P.S.
A small suggestion about your chart: remove the "because it can't be indexed" phrase, because the fact that a resource can't be indexed has nothing to to with the assignment or passing of PageRank. For example, you can create a NOINDEX resource and assign PageRank to it. That resource, by default, will also pass PageRank to other linked resources. So in my opinion you shouldn't create this connection between PageRank flow and indexing, because the two things are not really related.

First, we should all remember that the first version of PageRank was a (very) rough way to estimate the probability of each page on the web to be reached by a web surfer. The PageRank formula was designed taking into account how many links each page contained because knowing the quantity of those links was necessary to estimate the probability of each link to be clicked by the user.

Now, reality tells us that as long as a user can see and click a link, that link existence in the page will decrease the other same-page links' probability of being clicked.

When Google introduced the rel=nofollow attribute, WhateverRank was calculated so that the total amount of WhateverRank sent by a page was divided between all the do-follow links in the page. This was a big mathematical mistake because doing things in this way created an artificial WhateverRank boost for the resources linked by the do-follow links. It was artificial because the presence of a rel=nofollow attribute in a link doesn't change at all the actual probability of the remaining links in the page to be clicked by users.

To fix their mistake, Google changed the formula so that the WhateverRank sent by a page was divided between all the links it contained, regardless of their follow/nofollow status. This update made the WhateverRank calculations able to provide values that matched in a better way what users can actually do while navigating web pages.

The lesson that Google learnt from this mistake was that a formula like that should always take into consideration what happens in real life.

Now, how should a formula like that behave when one of the two links in a page points to a resource that sends a 404/410/403/4xx status code? In my opinion +Trey Collier has the correct answer: if the Google engineers have learnt a good general lesson from the nofollow mistake, we should expect that the link that points to the reachable resource would get about half of the WhateverRank owned by the page. The user sees two links, so the probability of each of them to be clicked is roughly 50%.

(By the way: that 50% figure assumes that WhateverRank is divided evenly among all the links of a page, which probably it's not what the search engine actually does, but that's a different topic)

Now we have a good answer to the question "what happens to the WhateverRank transferred by the links that point to the reachable resources". It's a good answer because:

1) This kind of approach/formula models in a better way what actually happens in reality: as long as a user can see and click a link, that link contributes to the probability of the other links to be clicked

2) It's the correct mathematical way to approach the situation

3) Google already made in the past the mistake of artificially boosting PageRank. From that specific mistake, they should have learnt a general lesson about WhateverRank distribution 

So, we don't have an official answer by Google, but all the signals point to the same direction: all the links in a page should be taken into consideration in order to better calculate a distribution of click/visit probability.

If you are interested in WhateverRank distribution, I would suggest to focus more on how much each link is visible to the user, because we should assume that dividing WhateverRank  _evenly_ between all the links in a page, regardless of their actual probability of being seen and clicked by users, is not a very good way to design a probability-estimation formula.

Cheers!

¹ I've called it "WhateverRank" just because I think it's highly unlikely that Google hasn't developed something more sophisticated of that old goat in the last 17 years.

Can we please have a direct answer on the fate of PR associated to the link from A the 404ed-D in Gary's pretty graph?

Does it goes to the unicorns (as a nofollow-ed link juice does in my understanding, so far I've always assumed this way), or is the link discarded before redistributing PR, or what?

Of course, a "classified" answer is understandable, but if no secret-sauce is involved, the direct answer to the point would help us taking better decisions in some scenarios.

Apart from revealing changes to how internal NOFOLLOW links are handled (which wasn't part of original patent), Google has never publicly explained how they handle links to dead pages in terms of PR distribution AFAIK.

If you can point me to any references from original patent covering this I'd like to give it a gander again.  

+Alistair Lattimore  and I have alluded to PR from a page is divided equally to all the links that are followable……and that getting rid of 404 links will increase the left over links with a boost of PR link Juice.    Google's patent on this (PageRank) explains it quite well.    

Google will not deny or confirm how they treat links to 404 pages.    But their Patent does….and a page that doesn't exist can NOT hold or receive any PR.      Am I missing something here?

The 2 obvious choices are to sink the PR passed over a 404 or remove that link from the PR distribution, and therefore redistribute the PR over the remaining links from the document.

I've been unable to find anything official on this, and perhaps it's coming close to the secret sauce I mentioned on your post.

Google have confirmed in the past that they changed how they calculated the PageRank for outlinks with regard to PR sculpting. Last word on this is they calculate how much PR should flow through all outlinks, any links determined to not flow PR (rel=nofollow as an example) – the PR goes the way of the unicorn.

Google have previously stated and Gary reconfirmed above that error documents can't maintain PR in Google's systems.

In your example above, my expectation is that by removing the links pointing to 404 pages – with fewer outlinks per page, each link that does receive and flow PR will get slightly more.

On the robots.txt – put a nice big iron gate across the path

I've learnt something new also – I hadn't realised case 2, and always felt that if enough links pointed at something behind robots.txt it could get indexed.  I had always assumed the login pages of some of the bigger sites I've worked on got indexed that way, but obviously they must have been indexed previously.

Thanks for input both +Gary Illyes and +Pedro Dias (I'll dream of unicorms shortly sigh)

As for the roboted (disallowed by robots.txt) pages, you have two cases:
1. the page became roboted after it was indexed, in which case we may decide to keep the page in our index with all its accumulated signals, including PageRank. Since we can't access the page any more, it will not pass PageRank
2. if we find a roboted page and it's not already indexed, we will not index it and thus can't get or pass PageRank.
Not quite sure how to put this on the diagram though. 

Does this help?

BTW one other scenario that might be nice for add there is links to URIs that are inaccessible due to robots blocks.  I always found that to be an interesting issue.

FWIW I'm looking for justification to get a large site to clean up non-trivial volumes of 404s, but cannot play the uX card. Has to be SEO driven.

Hope this helps with your research