Thank you so very much for the detailed reply. Much to ponder.

When guessing, “the first link listed may hold more weight,” I was thinking not so much logically, as by analogy, which betrays my classical studies training. First, I noted that search engines list links in SERPs from (they hope) most to least relevant. Second, when giving a list of links, people are savvy to the fact that the first link attracts the most clicks, so they may list the one they consider most important first. Third, as link lists get updated, newer links tend to get added to the bottom — not always, but it’s the easiest way to edit and add new links. For all those reasons, I could see that search engines might give a little more weight to the first link in a series.

But that was a flailing guess based on only one isolated case. You’re right; I’d need a lot more testing to prove it. And I haven’t really done enough to be sure about the boldface. Last week, I tried switching the boldface from one test page to the other in each of my doublets. Google’s seen the change but has not switched them in the SERPs. Drat.

The trouble is, this really takes a heck of a lot of time to do properly; nor am I particularly good at this kind of experimentation. I knew it would be difficult to isolate all factors except the one I was trying to test; it’s even harder than I thought! It’s so tempting just to assume that you and Matt Cutts and the other Experts know better than I. It is a lot easier for me to play textual critic, comparing and analyzing your posts and Matt Cutts’ sound bites et alia like a classicist trying to sort between the skewed accounts of the Catalinarian Conspiracy in Cicero and Sallust. (In fact, it seems to me that even real SEO experts do their share of literary criticism rather than scientific testing, trying to interpret what Matt Cutt means or reading other sorts of verbal tea leaves).
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I will keep puttering away at these experiments, but I fear at this rate it will take a year before I’ve compiled enough meaningful data on even the simplest of factors (boldface). By that time, of course, every major search engine will have changed its algorithm.

I’m always about one step from deleting this blog as bogus information, but hopefully I’ve made it clear that I am an amateur trying to understand SEO, rather than a professional.

Before you form that hypothesis, ask yourself how many ways a particular outcome of an experiment might be explained.

For example, your conclusion for experiment 1 says: “This doesn’t disprove that bold has any significance; it merely suggests that links have more weight (unsurprisingly).”

Agreed. It seems to suggest that links have more weight than bold-face — but I can think of several ways to challenge that conclusion as you don’t seem to have isolated repetition as a potential factor.

However, a little above your experiment’s conclusion you draw another inference: “From this I conclude that the first link on a page gives more of an SEO boost than following links.”

Is there any logical reason for this to be so? If you think there is, can you articulate it? If you cannot develop and articulate a logical explanation then, lacking a hypothesis, your conclusion is irrelevant (to the facts).

You should be able to look at the behavior and infer some sort of explanation (even though it may be wrong). That inferred explanation should then lead you to more testing.

I can easily show that first links on pages don’t have any SEO value — I have pages on several of my own sites where the anchor text for the first links is not being passed to other pages, and yet anchor text from links further down the page IS being passed.

If it were really as simple as “the first link counts the most”, then there should be no secondary links that pass more value than first links — OR we should have an explanation for why there are exceptions to the apparent rule.

In other words, in science one rarely comes up with a hypothesis that terminates a line of investigation. Search indexes are evolving mechanisms. Their behaviors are constantly changing.

One of the most unprovable hypotheses in science is the hypothesis that an evolving system exists in an unevolving state. That is, maybe at one time only the first link counted but now it doesn’t; or maybe at one time other links counted more but now the first link counts the most; or maybe something else is happening now but at some time in the past one of the two prior claims was true; or maybe all the claims have been true at one or more times in the past and will be true at one or more times in the future.

In fact, it can easily be shown that there is no statistical significance to differentiating PageRank-like value between links on a page but that there may other significant aspects to link order on a page (such as the order in which the links are passed into crawl queues).