Ideally I would like to generalize this to a wider (but also degenerate) set of matrices, but so far this is the only conjectural working example that I can give for the moment:

Let the $N$ by $N$ lower triangular matrix $L$ be defined by the tetration:

$$\text{If } \text{ mod}(n,tk)=0 \text{ then } L(n,k)= \underbrace{e^{e^{\cdot^{\cdot^{e^{1}}}}}}_m \text{ else } L(n,k)=0$$
where:

$$n=1..N$$
$$k=1..N$$.

and let the $N$ by $N$ upper triangular matrix $U$ be defined as:

$$\text{If } \text{ mod}(k,tn)=0 \text{ then } U(n,k)= \mu(n) \text{ else } U(n,k)=0$$
where:

$$n=1..N$$
$$k=1..N$$.

and let the square $N$ by $N$ matrix $A=L.U$ when $t=1$,

and let the square $N$ by $N$ matrix $B=L.U$ when $t=2$,

and let the square $N$ by $N$ matrix $C=A+B$.

Denote the eigenvalues of $A$ with $\lambda_A(n)$ and

denote the eigenvalues of $B$ with $\lambda_B(n)$ and

denote the eigenvalues of $C$ with $\lambda_C(n)$.

It then appears that for $m$ pieces of natural logarithms of the eigenvalues $\lambda_A(n),\lambda_B(n)$ and $\lambda_C(n)$, the following holds:

$$\text{sgn}(\lambda_A(n))\underbrace{\log(\log(...\log(}_m |\lambda_A(n)|)...))+\text{sgn}(\lambda_B(n))\underbrace{\log(\log(...\log(}_m |\lambda_B(n)|)...))=\text{sgn}(\lambda_C(n))\underbrace{\log(\log(...\log(}_m |\lambda_C(n)|)...))$$
as $m \rightarrow \infty$.

Associated Mathematica program to demonstrate the conjecture:

```
(*start*)
(*Mathematica 8.0.1*)
Clear[C1, C2, A, B, s, nn, i];
mm = 32;
"Counts to 32"
t = 1;
Monitor[Table[
A = Table[
Table[If[Mod[n, t*k] == 0, Exp[Exp[Exp[1]]], 0], {k, 1, nn}], {n,
1, nn}].Table[
Table[If[Mod[k, t*n] == 0, MoebiusMu[n], 0], {k, 1, nn}], {n, 1,
nn}];
a = Eigenvalues[A];
N[Sum[Sign[a[[i]]] If[a[[i]] == 0, 0,
Log[Log[Log[Abs[a[[i]]]]]]], {i, 1, nn}], 6], {nn, 1, mm}], nn]
sumA = Round[%]
"Counts to 32"
t = 2;
Monitor[Table[
B = Table[
Table[If[Mod[n, t*k] == 0, Exp[Exp[Exp[1]]], 0], {k, 1, nn}], {n,
1, nn}].Table[
Table[If[Mod[k, t*n] == 0, MoebiusMu[n], 0], {k, 1, nn}], {n, 1,
nn}];
b = Eigenvalues[B];
N[Sum[Sign[b[[i]]] If[b[[i]] == 0, 0,
Log[Log[Log[Abs[b[[i]]]]]]], {i, 1, nn}], 6], {nn, 1, mm}], nn]
sumB = Round[%]
"Counts to 32"
C1 = A + B;
Monitor[Table[C2 = Table[Table[C1[[n, k]], {k, 1, nn}], {n, 1, nn}];
c = Eigenvalues[C2];
N[Sum[Sign[c[[i]]] If[c[[i]] == 0, 0,
Log[Log[Log[Abs[c[[i]]]]]]], {i, 1, nn}], 6], {nn, 1, mm}], nn]
sumC1 = Round[%]
sumA + sumB
Sum[Table[
Sum[If[Mod[n, k] == 0, MoebiusMu[n/k], 0], {n, 1, nn}], {nn, 1,
mm}], {k, 1, t}]
%%% - %%
%%% - %%
t = 1;
muA = Table[
Sum[If[Mod[n, t] == 0, MoebiusMu[n/t], 0], {n, 1, nn}], {nn, 1, mm}]
t = 2;
muB = Table[
Sum[If[Mod[n, t] == 0, MoebiusMu[n/t], 0], {n, 1, nn}], {nn, 1, mm}]
muC = muA + muB
(*end*)
```