yes, the mystery of the secret name . It involves changing the Bjornaer it doesn't actually target the AC's.
This question came up at Grand Tribunal 2007, and Mark Shirley gave what I thought was a really good answer. I'll see if I can do it justice.
Arcane Connections are formed according to two laws: the Law of Sympathy and the Law of Contagion. That is, they either mystically resonate with the target, or they were once part of the target. The latter case is primarily what we're concerned with here, and a common example is a hair from a person's head. A hair from my head was once part of me, so it is an Arcane Connection to me because of the Law of Contagion. I was never part of the hair that came from my head, so by that same Law I am not an Arcane Connection to it.
I believe it was Nathan Hook who brought up the question of a stone that is split in half: are the two halves Arcane Connections to each other? There's probably some wiggle room here, but the simple answer is no, because each half was never part of the other half. Each piece would be an Arcane Connection to the stone, but where is the stone?
...
(Gonna have to think about that one...) 
Hi,
That's one way to look at it; another is that both were once part of the same whole, and are therefore ACs to each other. "Once part, always part."
This avoids the following:
Because the stone is where it always has been, still whole. Once part, always part.
That said, I don't believe that if A is an AC to B then B must be an AC to A.
Let's consider an apple that falls from a tree.
This apple will always have come from that tree. In many ways, coming from that tree is the most important thing about that apple, because if it would not exist if not for the tree! If it came from a different tree, it might be an orange rather than an apple. So if you have the apple, you have something extremely connected to the tree.
The tree, however, puts forth many apples. Many of them fall, or are eaten, or rot, and the tree continues as it was. The apple was once part of the tree, but not a very important part. The connection is therefore similarly trivial.
Anyway,
Ken
Cheers Erik - for putting it into a nice perspective, whether your words or another's thoughts or vice versa (I wanted one of those shocks too).
I came to think of a possible philosophical pursuit of magi -might even be a life-long rivalry between two grizzled magi- on how to, and with what means, to achieve a fabled "Golden Median" by which you can divide what was whole in such a way that it in essence stays none the part of the other yet both the sibling of its mirror part - both being an Arcane Connection to the other. I could even think of various possible magical items crafted by a Verditii Master...
Maybe we could reach a compromise on this:
An AC works both way if the thing it is connected to cannot grant further AC of the same type. Thus, it still "misses" the AC.
Examples include: Bodyparts of animals and humans as these do not regrow, the head of a stone statue, a stone split in half.
It excludes: A strand of hair or feather, an unrecognizeable small piece of a statue, a splinter of some stone (you can produce many more of them and the stone is essentially the same).
This does leave some things undefined but the SG or troup can rule on these when they come up to generate the best story.
However, if severed parts do regrow due to magic that normally do not, the AC becomes one-way, as the target is no longer missing that part.
ACs due to the Law of Sympathy are always only one-way ACs.
You did better than me! I was preparing an answer when this came in; and you expressed the rule I use far more succinctly than I was going to!
Cheers.
Mark
(friggin' closet Jerbiton.
[shakes fist at heavens]
There will be NO COMPROMISE!)
(Ok, thought about it. Here we go...)
[color=darkred](Note - The Rules stem from the Laws, but reinterpret them to fit the Game, re balance etc. Readers should keep this always in mind, that there is a distinction between the Laws, and the Rules that spring from them. If there is a contradiction, StoryGuides should use their common sense, or just follow the Rules, depending on your own essential nature.) 
The Law of Sympathy is mainly one of symbolism. A feather can represent a bird (any bird), and a stone can represent a castle (if poorly.) Voodoo dolls and such are another classic example. When actually made from an object that came from the thing they represent, the Law of Contagion adds strength to that representation.
The Law of Contagion is, indeed, what we are mainly addressing here. These Laws are not included in the Rules, and so it seems the original wording of the law has become generalized.
Not quite. You're using the wording of the rules to rephrase the Law, which then changes the reading of the rules.
At its simplest, the Law of Contagion is: "Once together, always together." At length, it can be expanded to imply that when two things are in contact, there exists an astral bond between them that is not lost when they are separated, such that actions taken on one can effect the other.
It has nothing to do with Parts subservient to Wholes, nor Platonic Philosophy, unless that is an Ars re-interpretation (which may be a valid and necessary evil. If that's so, then never mind.)
The Law of Contagion says that if the two halves of the rock were touching, especially if they were connected, and (what's stronger than "especially"? ) extra-double-especially if they were one, then when split in half, they are still connected to each other.
Now, as written, the Rules do not emphasize "contact" or touching, but "an association", which is a partial extension of "Once together, always together."* The two halves of the rock are, and always will be, directly associated with each other, and so are now AC's to each other, despite the original "whole rock" not existing in the philophical sense.
(* So, a favorite toy is an AC to a child, even tho' they were never part of a single discrete, permanently connected thing.)
(Now, I found an interesting quibble, that says that if hair is cut from a person, and then that person cuts their hair again, that first part loses it's connection to the person, because it was only "together" with the now-missing intermediary piece. By this interpretation, one would simply have to polish the broken surface of one half of the stone to break the connection. But I'd submit that the association still exists, despite the technicality, from the Law of Sympathy, if not narrowly defined Contagion.)
So, we look at the simplest wording of the Law of Contagion, "Once together, always together", and ask ourselves: Was the magi ever "together" with their hair? Simplest answer: Yes.
Once together, always together. (Rules about degrading over time notwithstanding.)
(Getting back to the original question, were the original individual pawns of vis ever "together" with each other? Tougher question, especially with an eye to the "association" issue of the rules. I'd say "no", no more than any two neighbors are "together", in this sense.)
Now, the Rules, for game balance, often do reinterpret or modify such absolute laws, and for various good reasons. And SG's should always do the same, with a mind to their own sagas and ongoing stories, short and long-term. But, in the absence of such canon or individual interpretation, and keeping in mind the Rules vs. the Law, I'd say it looks like a two way street.
We must be sure that everyone understands that the Laws of Contagion, Sympathy and so forth are not authentically medieval. Some of the concepts are medieval, but the laws were formulated by Frazer (Similarity & Contagion) and Mauss (Antipathy & Sympathy) at the turn of the 20th century.
The concepts occur in various translations of The Book of Secrets by "Albertus Magnus". Whereas Albertus Magnus was authentic C13, his so-called books were not; they were first written in the C15-C16. Most of my interpretation/terminology here is from The Book of Secrets.
That's not Sympathy, according to "Albertus". That's a mixture of Similarity - similia similibus, or "like to like" (that is, the effect resembles its cause) - and Affinity (a contagious object has affinity with the thing that its 'parent' represents). For instance, a voodoo doll (or 'poppet', in medieval terminology) is Similar to the person it represents. A signature is not Similar to the person who scribed it, but has an Affinity to him and thus can represent him.
In the C19, mythographers created the uber-law of Sympathy from Similarity and Affinity.
Where as I (somewhat predictably) would say "yes" (and I'm not trying to be contrary, honest!). An Affinal Arcane Connection would exist between vis and its source - one which would decay faster than a Contagious one. I'd also say there would be problems fixing such an Arcane Connection to prevent the decay - how do you infuse a pawn of Vim vis into a pawn of (say) Ignem? What sorts of merriment or explosions could result? 
All IMO (just in case I haven't said that enough...)
Cheers,
Mark
Ah- well, in that case my whole premise may be flawed, as I acknowledged in my aside re Ars terms. It seems that the wording varies from source to source, and I had not found the primary source.
Well, for lack of a better term. Given certain other RPG associations, perhaps not the best choice of words. 
As for the rest, it seems we're starting at two different points, re the exact wording of The Law. From pars pro toto, I can readily see your position (tho' I do not, quite yet, relinquish my own...)
It may have been Isaac Bonewits. He got a PhD in Magic 
or somesuch nonsense, and part of his thesis was made into a game supplement in the 19[7,8]0s. GURPS reprinted this as "Authentic Thaumaturgy" - a deeply annoying* but somewhat insightful book, IMO. He came up with lots of 'Laws of Magic' and put snappy mnemonics to them. "Once together, always together" sounds like one of his.
*such as the constant bashing of scientists who he considers agents of "The Man", diatribes against organised religion, and his insistance that magic is real because he personally has cast spells which work. I had to make three attempts to finish the book because I can't stomach statements of any of these three types. YMMV.
[color=darkred]EDIT: Frazer used "Once together, always together" in The Golden Bough, so it goes back further than Bonewits
Mark
Mark,
By your reasoning, take a symmetrical brick split in half--would they be an affinal arcane connection by having some properties of each other?
Similarly, would my left pinky-toe be an arcane connection to my right pinky-toe?
-Mario
Yes, but it wouldn't last very long in my saga. Perhaps in the Hours or Days category.
That's a more difficult one. One toe has acquired nothing from the other, they've probably never been in contact! Each toe has acquired properties of MarioCerame, but the right toe has no affinity for the left toe. So, probably not.
Having said that, if it was dramatically necessary for my saga, I'd possibly allow it, with a duration of Hours. I never let theory get in the way of a good story!
Mark
He probably went to City College with Peter Parker, noted for his degree in Science.
That, or Miskatonic U. (Go 'Pods!)
Altho', I recall a classmate, split Philosophy/Chemistry Major, doing his undergrad thesis on Alchemy. Truth!
I didn't thought such a thing were possible. This is so... What did he intend to become? An alchemist???
Fantasy writer?
As I recall, the thesis spanned the two disciplines, and pursued different approaches to the nature and categorization of matter, partially from a historical but more from a modern perspective. For one, he re-presented the periodic table based on progressions other than atomic weight, and discussed the implications thereof. (I've seen that once, and you get some ~very~ interesting patterns ("trends") of elements when you base it off of other criteria.)
That is, basically, the thesis presented alternatives to the standard Metals/Nonmetals, the Metalloids, Alkali metals, Alkaline earth metals, Transition elements, Halogens, Noble gases and so forth, and raised questions about assumptions and generally perceived patterns, the blindly accepted paradigm that stifles creativity. I think he used some recent "breakthroughs" to show, after the fact, that a different view may have led to the same conclusions earlier.
When you don't have metals and non-metals, what DO you have?
Bright guy. He went into Chemistry and research, iirc. (No money in Philosophy!)