Question about tetration methods

[Gottfried]

My head gets fuzzy thinking about it for Kneser though, it just seems unnatural to me.

Yes, perhaps it is too fuzzy, and "eats up" the joy of discovery...                

One point when it became too much for me was the procedere of Ecalle, when it went to involve the logarithm of the Schroeder-function (that's simple enough), but then to use the reciprocal (which might still be doable/expressible though) and then including the integral term. It seemed to me that to express this with Carleman-matrices needed the extension to negative indexes, so to introduce the ability to work with Laurent series or even fully 2-way-infinite series \( \sum_{k=-\infty}^{+\infty} a_k x^k \). I've seen, as I mentioned earlier, that Eri Jabotinsky worked with such an extension, but I gave up for my part... (feeling dried out, natural process, simply).

So also for me, one might say, the "carlemanization" of the analysis of the tetration-function became unnatural somehow/somewhere, and since the Ecalle-result/-process seems to be an important and a good one I didn't try to go further, ... and don't make bold statements since :-) ...

Though, there definitely exists Carlemann matrices which solve the equation--can't imagine the construction method though.

I agree, with the words above. Only that there remains still a tickling of curiosity, how ... perhaps ...