Galaxy array notation

Galaxy array notation is a notation to generate large numbers created by Googology Wiki user Binary198. The creator essentially insisted the well-definedness through the creation of the article of Tetra-trideca including the statement "Tetra-trideca is the largest number which Binary198 has coined using Dimensional Galaxy Array Notation.", which should imply that Galaxy array notation is well-defined up to that realm. However, the original source of the number and the notation is a personal website which is not peer-reviewed, and hence the information was incorrect. Namely, the notation itself is ill-defined as we will explain in #Issues.

Definition

Here is a copy of the original definition:

Linear and basic rules

Case M1 (2 entries) {a,b}=ab

Case M2 (tailing zero) {#0}={#}

Case M3 (replacement) {a,b#0,d+1}={a,b#b,d}

Case M4 (expansion) {a,b+1#c+1,d} = {a,{a,{…{a,a#c,d}…}#c,d}#c,d} w/ b layers

Separators

Case A1 (semicolon) {#x;y#}={#x,x,…,x#} w/ y copies of x

Case A2 (curly braces) {#x{0}y#}={#x;y#}

Case A3 (curly braces) {#x{n+1}y#}={#x{n}x{n}…{n}x#} w/ y copies of x

Issues

Similar to the creator's old notation Hyper-Robin notation, it has many issues in the formality:

1. The lack of the definition of the domain: The domain of the notation is not clarified. Therefore it is ambiguous what expression can be evaluated. Therefore we need to ambiguously guess that the domain is some uncertain subset of the set of formal strings.
2. The lack of the precise quantification of variables: The variables a, b, #, c, d, x, y, and n are not appropriately quantified with specific range. Therefore it is ambiguous what conditions are assumed in each computation rule. Therefore we need to ambiguously guess that those are either an integer or a valid expression, which is also undefined.
   1. Since "b layers" makes sense only when b is a positive integer, it is good to guess that at least b is supposed to be a positive integer rather than a valid expression of the notation.
   2. If we allow c to be negative, we can apply both M3 and M4 to {2,2#0,2}. Therefore c is perhaps intended to be always non-negative.
3. Ambiguous operator: Since the application of "+1" is not defined for a valid expression, it is good to guess that the occurrence of "+1" in the rule means an addition of 1 to integers or just a formal string of length 2. This problem occurs due to the lacks of the definition of the domain and the precise quantification of variables.
4. Invalid expresion: The result of the application of M3 to {2,2,0,1} is {2,2,2,0}, and the result of the application of M2 to {2,2,2,0} is {2,2,2,}. However, since there is no rule applicable to {2,2,2,}, the computation refers to an invalid expression.
5. Incomplete partial specialisation:
   1. There is no rule applicable to {1;2;3;4}. We note that a single symbol like # in a single equation should stands for a common value, and hence A1 is not applicable to this expression.
   2. There is no rule applicable to {10{0}10,1}. By the same reason above, A2 is not applicable to this expression.
   3. There is no rule applicable to {10{10}10{10}10}. By the same reason above, A3 is not applicable to this expression.
6. Overlapping partial specialisation: both of M4 and A1 are applicable to {1,3;3,1}, and the resulting expression is not unique.

As a result, the notation is ill-defined.