Thank you very much for the comments, Mike and Rua. Here is my reply. It has quite a lot equations, so I wrote it in Latex. I can send you (and other people who are intested) the Pdf by email. Please contact me yu.ernest.liu@hotmail.com
Anyhow, I paste the ".tex" here (which is a bit hard to read).


Now I do realise that there exist those counter-examples of the hypothesis. I also realised why it happens. By adding a little bit stronger condition (besides all of the 6 conditions matrix A should satisfy), the main claim $A^p {\bf N}= R{\bf N}$ should still hold. After all, the observation $A^p {\bf N}= R{\bf N}$ is still interesting.

I first explain the counterexamples. One counterexample is that (as you suggested), let
\[
B = \left(\begin{array}{ccccc}0 & 0 & 0 & 0 & 1 \\1 & 0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 & 0 \\0 & 0 & 0 & 1 & 0\end{array}\right) \text{, _then}
A = \left(\begin{array}{ccccc}0 & 0 & 0 & 1 & 1 \\1 & 0 & 0 & 0 & 1 \\1 & 1 & 0 & 0 & 0 \\0 & 1 & 1 & 0 & 0 \\0 & 0 & 1 & 1 & 0\end{array}\right)
\]
So A satisfies all the 6 conditions and $A^2 {\bf N}= 4{\bf N}$, but the eigenvalues of A are 2, $-0.5 \pm 1.54i$ and $-0.5 \pm 0.36i$, respectively.

When I try to construct an analysable chemically realistic self-replicating system (based on the principles described on page 4) whose recurrence matrix is A, I cannot succeed. The reason is that, in order to have the recurrence matrix as A, the corresponding chemical system must be written as something like:
\begin{equation*} \begin{split}
C \rightarrow D + E \\
D \rightarrow E + F \\
E \rightarrow F + G \\
F \rightarrow C + G \\
G \rightarrow C + D
\end{split} \end{equation*}
One cannot find proper integers to replace $C, D, E, F$ and $G$, to satisfy the principles to construct an analysable chemically realistic self-replicating system. Because one cannot find positive integers to satisfy $C = D + E, D = E + F, E = F + G, F = C + G, G = C + D$, as you see, if you add all the 5 equations, you get $0 = C+D+E+F+G$. One necessary condition to be chemically realistic is that ``for any chemically realistic self-replicating system, there is at least one type of resource molecule (in the middle on page 4), which the system above cannot satisfy. That is to say, {\bf there is no chemically realistic self-replicating system corresponding to the recurrence matrix A shown above}.


Going back to the hypothesis, the motivation I put forward this hypothesis is that, for all the analysable chemically realistic self-replicating systems I investigated till size 7, I observed that if $A^p {\bf N}= R{\bf N}$ then $|\lambda_1| = |\lambda_2| = ... |\lambda_u|= \sqrt[p]{R}$ and $\lambda_{u+1} = \lambda_{u+2} = ... \lambda_{m} = 0$ (let's denoted this two equations as Eq (1) for now). So the hypothesis should be
\begin{equation*} \begin{split}
& \text{All analysable chemically realistic self-replicating systems up to size infinite} \\
\Rightarrow & \text{Eq (1)}
\end{split} \end{equation*}
And I was thinking (for which now I realised that it is wrong),
\begin{equation*} \begin{split}
& \text{All analysable chemically realistic self-replicating systems up to size infinite} \\
\iff & \text{All the 6 conditions (Wrong!!!)}
\end{split} \end{equation*}
But instead, it should be
\begin{equation*} \begin{split}
& \text{All analysable chemically realistic self-replicating systems up to size infinite} \\
\iff & \text{All the 6 conditions + other conditions}
\end{split} \end{equation*}
That is, all the 6 conditions are only necessary conditions, but not sufficient.

So, matrix A (including other similar counterexamples, such as 5-cycle 0/1 permutation matrix B) satisfies all the 6 conditions, but not corresponds to any analysable chemically realistic self-replicating system.

The ultimate way to save this hypothesis is find all the other conditions, which I have not figured out. But the one that has to be added is that
\[
\text{For A, the columns cannot all have two non-zero entries.}
\]
Because ``for any chemically realistic self-replicating system, there is at least one type of resource molecule (in the middle on page 4).

{\bf This extra condition can rescue the hypothesis from this type of counterexamples. But I am not sure whether there are other type of counterexamples.}