Let us consider the following 3 rules:
rule 1: those systems where each chamber definition must contain three different letters. All of these systems satisfy the closure principle of manufacturing the agents that convert or manufacture other entities.
rule 2: those systems where the letters present in the first chamber can be used to generate letters in other chambers forming a chain reaction until all the reactions have taken place.
rule 3: those systems where no two chambers contain the same two letters.
Rule 2 and rule 3 are extra constraints on rule 1. Systems that satisfy either rule 2 or rule 3 also satisfy rule 1.
It should be noted that there are systems with selfentailing boxes that are not constrained by rule 1.
3 Chamber Systems
Let us review how the 3chamber system was chosen. Consider all they ways to pick the agents that convert the following inputs to outputs.
:a>b  :b>c  :c>d  from  {b,c,d} 
Applying rule 1, there is only one choice.
c :a>b  d :b>c  b :c>d 
This system also happens to satisfy rule 2 because c:a>b has letters b and c present that could be used in reaction b:c>d. Then letters d and b are present that could be used in reaction d:b>c.
Satisfying rule 2 (chain reaction) 
It does not satisfy rule 3 because as can be seen below (with the yellow highlights on b and c) there exists some pairs of letters that show up in more than one chamber. No systems satisfying rule 2 also satisfy rule 3.
c:a>b  d:b>c  b:c>d 
4 chamber Systems
Consider a 4 chamber system with all possibilities of picking the converting agents.
:a>b  :b>c  :c>d  :d>e  from  {b,c,d,e} 
If we apply rule 1, we get the following three possibilities as depicted in each row:
c:a>b  d:b>c  e:c>d  b:d>e 
d:a>b  e:b>c  b:c>d  c:d>e 
e:a>b  d:b>c  b:c>d  c:d>e 
None of these systems satisfy rule 2. For example, in looking at the first chamber of the top row, consider letters b and c. In no other chamber in the top row does b operate on c to allow the chain reaction to continue. A similar thing can be seen for the letters applicable for the other rows as well.
Additionally, none of these systems satisfy rule 3. For instance, consider chambers with both letters b and c. All three systems contain 2 chambers with these highlights. There are duplicates of other letter pairs as well.
c:a>b  d:b>c  e:c>d  b:d>e 
d:a>b  e:b>c  b:c>d  c:d>e 
e:a>b  d:b>c  b:c>d  c:d>e 
Computer Program
With the aid of a computer program, systems of higher numbers of chambers can be tested for adherence to rule 1, rule 2, and rule 3.
Population 1 those systems satisfying rule 1
Population 2 those systems satisfying rule 2
Population 3 those systems satisfying rule 3
The chart below shows the results of running a computer program to test for and count the members in each population for systems with up to 12 chambers. Empty cells represent the value zero.

Click here for the source code. Save the file as Rosen.java and compile with the Java Development Kit downloadable from http://java.sun.com. Usage: java Rosen [start] [end] [pop] [print] where [start] = starting component number [end] = ending component number [pop] = which population to determine [print] = whether or not to print out the various systems The printout will also indicate the number in each population for each chamber count. For example, java Rosen 7 7 2 y will produce 7: e f g h b c d 7 1 which are the agents selected from the possible choices for population 2 of a 7chamber system. 
** The calculations required for (N=13, population 1) were so extensive that the program was aborted before completion.
The next page will show the two seven chamber diagrams from population 3.