This is why, again, I think the beta methods all produce the same tetration. Choose a strip \( \mathcal{S}_{a,b} = \{ s \in \mathbb{C} | a \le \Im(s) \le b\} \). Suppose we have a family of functions \( f \in \mathcal{B} \), such for all \( [a,b] \) there exists an \( X \) such that:
\(
\log f(s+1) - f(s) = 0 \,\,\text{as}\,\,\Re(s) \to \infty\\
\)
And \( f(s) \) is holomorphic for \( \mathcal{S}_{a,b} \cap \{\Re(s) > X\} \); such we can topologically assign,
\(
\log f(\infty) - f(\infty) = 0\\
\)
I believe that all prospects which start from this hypothesis all produce \( \text{tet}_\beta \). Which means,
\(
\text{tet}_\beta(s + x_f) = \log^{\circ n} f(s+n)\\
\)
Your function \( \text{Tom}_A(s) = \log \text{Tom}(s+1) = A(s) \text{Tom}(s) \) is of such \( f \).
Mind you, this is just a conjecture. But it's been numerically evaluated fairly well.
EDIT:
I also forgot to mention that as \( a,b \to \infty \) as \( \Re(s) \to \infty \) we still get \( f(a(s)), f(b(s)) \to \infty \). Which is the statement that on the boundaries it approaches infinity; and when we do the pull back this will construct a tetration with \( \lim_{\Im(s) \to \infty} \text{tet}(s) = \infty \). I mean to say that all such tetration functions should be identical. Kneser satisfies all the above conditions until this one I just added. Forgot to mention, non-normality at \( \Im(s) = \pm \infty \).
\(
\log f(s+1) - f(s) = 0 \,\,\text{as}\,\,\Re(s) \to \infty\\
\)
And \( f(s) \) is holomorphic for \( \mathcal{S}_{a,b} \cap \{\Re(s) > X\} \); such we can topologically assign,
\(
\log f(\infty) - f(\infty) = 0\\
\)
I believe that all prospects which start from this hypothesis all produce \( \text{tet}_\beta \). Which means,
\(
\text{tet}_\beta(s + x_f) = \log^{\circ n} f(s+n)\\
\)
Your function \( \text{Tom}_A(s) = \log \text{Tom}(s+1) = A(s) \text{Tom}(s) \) is of such \( f \).
Mind you, this is just a conjecture. But it's been numerically evaluated fairly well.
EDIT:
I also forgot to mention that as \( a,b \to \infty \) as \( \Re(s) \to \infty \) we still get \( f(a(s)), f(b(s)) \to \infty \). Which is the statement that on the boundaries it approaches infinity; and when we do the pull back this will construct a tetration with \( \lim_{\Im(s) \to \infty} \text{tet}(s) = \infty \). I mean to say that all such tetration functions should be identical. Kneser satisfies all the above conditions until this one I just added. Forgot to mention, non-normality at \( \Im(s) = \pm \infty \).