I apologize, I do not mean to berate your abilities. There is an aspect of the problem that you just seem to be missing. Let me try one more time.
My purpose is to model a very specific circumstance. That circumstance consists of a totally undefined set of information transcribed by a totally undefined process. The original undefined information is reality; the universe; everything. It is something I "know" absolutely nothing about (until I solve the problem, I can know nothing about it). However, there exists some information I do have available to work with. The problem is that the information I have available, though its source may be reality, has been transcribed by a totally undefined process (called my senses in my finished model).
The problem is totally analogous to the following. God creates a universe (providing the underlying undefined information) which can be absolutely anything. My job is to investigate the transcribed information available to me to determine what rules apply to the universe God has created. First, I argue that the very best I can do is to answer questions in a probabilistic manner. With a finite amount of information to analyze I clearly cannot be absolutely certain of anything. (I presume you understand and do not argue with my conclusion that the answer to any question can be posed in the form of the wave function notation).
Now we have the problem of the transcribing of the information. My view of reality is not what god created but is rather from the other side of this undefined conversion process. To fully understand the consequences of my having to deal with a totally undefined conversion process, let me create a scenario in your mental model of reality.
First, god creates the universe, a very specific set of information (the underlying rules of which I am to unravel). Then he takes 10,000 angels and gives each of them the job of encoding that set of information in their own secret code. Each angel can do what ever he wishes and is encouraged to bury the real data as well as he can: i.e., god wants to make the problem a difficult as possible on me. The only requirement on their code is that they must have a decoding procedure which will recover the original data from the transcribed result: i.e., the information about gods reality is there if you know how to decode it. (One can conclude from this that patterns in the original data will require patterns in the angel's transcription).
Now, my problem is that I am to uncover the rules of god's reality. I want to approach the problem with a method of analyzing the data which will lead to that result. If my approach yields a different result when I work with coded information from any given angel, then my answer is clearly wrong; the angel has won and his encoding has managed to hide the real rules.
So I ask myself, what kinds of things can the angel do which might change my data that I am working with from the data god created (which is not available to me). (I presume you will accept that the information, before and after transcription, can be seen a set of numbers). I say that, in order to keep my method of analyzing the data totally general (not dependent on anything any angel might do), it must yield exactly the same result no matter what encoding procedure any angel chooses.
Certainly adding a fixed number to any number in the original underlying set is so simple a code that I must include that as a possibility. Multiplying any number in the original underlying set by a fixed number is also possible. Remember, the angel can do anything he wishes so long as the original underlying set can be recovered (by the angel, not by me). Clearly, I can never hope to recover the original set, all I can hope to do is to find relations between patterns in the data. (But it is central that I must avoid patterns the angels can create on their own!)
That is, after a sufficient quantity of the angels transcription is analyzed, I can hope to find patterns in the data (to which I will attach names of my own creation) and discuss relationships between these patterns.
It is my position that, if I had the entire transcription available to analyze and if I were smart enough to figure it all out, repeated patterns in the angels transcription would correspond to repeated patterns in gods reality. I could then attach my names directly to those corresponding patterns in god's reality without ever knowing the exact patterns. (It is the existence of the patterns and the relations between them that is significant, not the actual specific patterns.) That is, I would have gotten around the angel's coding.
I will leave the issue at this point hoping I have cleared up the symmetry problem and why I am of the opinion that the asymmetry you refer to is not pertinent to the problem I am confronting.
Sorry again about upsetting you -- Dick