The error you are making is mentally trying to model what I am doing into your personal mental model of "how you solve problems". As I tried to explain to Harv a great number of times, I am not solving a problem! I am instead modeling the solution of the problem. I call it modeling an explanation because the solution of any problem is really an explanation: i.e., a method of obtaining expectations consistent with what is known.
It is very important here that all possible solutions are included. The model must be totally general or there might exist a possible solution which is not included within the capability of the model. If one is going to create a general model of a valid explanation (and valid means that the explanation is totally consistent with everything which is "known") which is based on some given information (which certainly cannot exceed "everything which is known") then you must be very careful not to exclude a possibility by assuming you already know some particular thing relevant to the problem.
I think you typed this line too quickly as it does not make grammatical sense: "My experience is that the elements of C is a function of the mapping procedure the mapping procedure." I am guessing that what you meant to say was: your experience is that the "interpretation" of the elements of C is a function of the mapping procedure. Please note that the elements of C are the elements of C. Part of solving the problem is deciding exactly what the elements of C are. That is something the solver must accomplish and is part and parcel of his explanation.
As I said in my presentation, a model of an explanation must posses two fundamental components: the information to be explained and the mechanism used to generate expectations for possible additional information. A short time later, I say these two parts must consist of a set of reference labels for the elements of A and an algorithm which will yield the probability of any specific set B derived from A. You should understand that providing that set of reference labels for the elements of A is exactly the act of deciding what the elements of C are in that explanation which is to be given. And certainly, the whole structure of your explanation is a function of that decision.
But there is a very important point here. The fact that you have solved the problem means that you have conceived of a set of labels (think in terms of "defining" the elements of B) and provided a mechanism for yielding those expectations consistent with C (and presumed consistent with A - though you can not prove that). But note that the mechanism must yield expectations consistent with C. That requirement cannot be a function your labels as they were invented by you and are not part of C (the information you have to work with).
To believe that the set of labels conceived of by you are the only possible set which will lead to a valid explanation is to presume your labeling procedure is the only possibility. In essence, that is equivalent to presuming that your explanation is the only possibility. You have to understand that I am working from a very abstract perspective. Just because you cannot conceive of a model which is independent of the mapping procedure does not mean one does not exist. All I am saying is that the correct model cannot depend on that mapping procedure. If it does, than the information necessary to produce that model is not contained in C, some part of it must arise through the mapping procedure: some part of your knowledge comes from things beyond the information available to you. I am presently trying to show you how to create a model which is independent of that mapping procedure.
With regard to your interpretation of C implied by your example of that pattern of photons, you are including a lot of information in your mental model of the problem (and thus in your expected explanation) which is not in your defined C: "the pattern of photons impinging on a detector". You say, "presumably A is the pattern that results in the infinite limit of detections". I am afraid A would have to be a lot more than that (or the phenomena you are attempting to explain are far less than you are presuming). By definition C consists of "all" information relevant to your explanation. Photons are pretty fundamental entities in your mental image of the universe and thus C becomes quite large. Now one element of C (a specific B) might consist of a pattern of photons impinging on a detector, but C itself would have to include enough information to give rise to a definition of "a photon", "a detector" and a sufficiently large collection of B's to decently present the problem.
And yes, C is the best that one can do by definition as C consists of all the relevant information available to you from which you are to create your explanation. Certainly, if your statement that "we are 100% sure that the path of the photons contained a double slit" is a valid measure of your expectations (totally consistent with all the B's in C) then C must include enough information to define exactly which B's in that collection of "all B's in C" you have labeled as validly having "a path", "a photon", and "a double slit" and (if you are actually 100% sure) there can exist no B's in all of C (all the information available to you) which could possibly be listed as a counter example.
From my dealings with Harv, I will not presume I have made anything clear with this exposition as I seem to fail quite regularly. Hopefully I am wrong in that expectation.
Have fun -- Dick