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close### Possible counterexamples to the "Hypothesis"

####
Posted by msteel
on
**
07 Aug 2018 at 20:57 GMT **

On page 13 of this paper, the authors put forward a conjectured 'Hypothesis'. A colleague (Rua Murray) and I attempted to prove this, but we came up against the following example which seems to provide a case where the Hypothesis as stated does not hold when m=5. Let B be the 0/1 permutation matrix corresponding to a 5-cycle, and let A=B+B^2. Then the 5x5 0/1 matrix A satisfies the six conditions listed earlier (p. 11/12) and it satisfies the condition A^p N= RN for p=2, R=4, and a non-zero 5x1 vector N (all 1's), yet the eigenvalue structure of A is not of the type suggested by the Hypothesis.

**No competing interests declared.**

### RE: Possible counterexamples to the "Hypothesis"

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ErnestLiu
replied to
msteel
on
**
12 Aug 2018 at 13:32 GMT **

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.}

**No competing interests declared.**