Sorry this has been so difficult. It is as hard for me to see what your problems are with what I say as it is for you to see what I am saying. The real problem is that we are not seeing the same problem. It has become very clear that you do not comprehend the problem I have attacked. Due to this, very fundamental misunderstanding, you probably won't understand my answers to your questions. If you go and look at the sequence of questions Harv asked (just below here) I think you will find the same difficulty arising there. He also does not seem to have any comprehension of what I am talking about.
Nevertheless, I will try to give succinct answers to your questions with the hope that it will clear things up a bit. First, I did not claim that PSI as derived from C was unique (in fact, my position is exactly the opposite of that). Remember, PSI is nothing more than a symbol standing for that most important second part of "the explanation": the algorithm which yields our expectations for any particular B. It should be very clear to you that the form and structure of PSI is very dependent upon the particular mapping chosen (that is, the meanings attached to those elements of A which happen to make up the B's in C).
It is the very fact that I can not presume that PSI or the mappings are unique which force me to work with the abstract perspective. As opposed to that, it is the standard position of the scientific community that the mappings (once again, that would be the meanings of the elements or concepts which go to make up their mental image of the problem) are absolutely unique. And, likewise, they believe that, if they do find an algorithm which provides them with valid expectations, that algorithm is also unique. Well, actually they will usually soften that claim a little by admitting that it is possible they are wrong, but you had better be able to show them their algorithm is invalid (find an incorrect expectation) before suggesting there are other "explanations".
The claim I make is quite different. What I say is that the expectations produced by the algorithm must be unique (this is because the B's which go to make up that C are unique, as they, in their entirety, constitute C: i.e., all the information available to us) and can not depend upon the meanings we have chosen to give to those elements of B. Note that C is not dependent on the mapping; C is whatever C is, (the total collection of information upon which the explanation is based)! It is rather the meanings attached to the elements of B which constitute the mapping.
Your question was, "can PSI ever be independent of mapping?" My answer to that question is I doubt it very much! However, if the explanation is to be valid, the expectations for any specific set B expressed in the specific mapping used to arrive at that PSI had better exactly correspond to the distribution of those B's in C because, THAT IS THE INFORMATION THE EXPLANATION IS BASED ON! The truth of this statement is beyond the presumptions of any meanings attached to the elements of B. It is true by definition; the definitions of A, B and C given in my paper.
You offered examples in which it appeared to you that [what C was seen to be, depended on the mapping]. Yes, what C is SEEN TO BE is exactly what the mapping is all about. The moment you apply a particular meaning to any element of any B, you are no longer looking at the abstract problem, you are instead looking for an explanation which applies to a specific given mapping.
You say that the most informative answer from me would be to suggest examples of C where the mapping was variable. Ok, take the problem of deciphering the ancient inscriptions from Knossos. See the web page:
To quote them, "There is no [consensus] on how to transliterate the Linear A symbols": i.e., the mappings (the meanings of the elements) used are still "variable".
Or, I should at least for your example of polling data, indicate how the data is independent of the rules used to collect it. Here you are clearly failing to include the entirety of C. Remember, C constitutes the entirety of the information relevant to the explanation to be discovered. From another perspective, you have already mapped 99.99% of the relevant information, essentially making the presumption that no alternate mapping is possible (that is, the mapping of the meanings of the terms you use to present the problem are not part of the problem).
Now, what you want me to do is to prove your presumption is false, essentially making the further assumption that, if I cannot prove your presumption is false, that fact alone constitutes a proof that your presumption is not only "not a presumption" but that it is unique: i.e., no other valid collection of meanings for the complete set of relevant elements exists. Maybe you are right, but you certainly cannot prove it. That is the fundamental problem of induction which philosophers have debated for centuries.
My point is, suppose we directly confront the issue. Let us see what we can say if we accept the fact that the mappings of those meanings can not be proved unique (we must then leave the meanings open). If we do that, we certainly do not know what we are talking about and must, of necessity, work in the abstract. So long as you attempt to avoid working in the abstract, you are avoiding coming to terms with the problem I am laying out before you.
I hope all that makes more sense to you!
Looking to hearing from you again -- Dick