Perhaps, I can clear things up here a little on my end since one of the paper's in question is my own.

First of all I apologize if this forum doesn't meet your standards, it is mostly dead now. Sheldon and I tend to be the few who actually post on here anymore. There isn't much action, so I rarely post here anymore, but occasionally something catches my eye.

First of all, my paper is numerically correct. If we want to define the hyper-operators for

and

, we need look no further than the recursive definition

where

If you want to work with tetration, it's simple. Just set

and there's your tetration, and a nifty formula for it. This I could prove in a heartbeat, just a coupla' lines of math. I extended this to arbitrary

and did so a little too swift and not without fault.

The paper that is available to the public does have gaps in its construction of the hyperoperators. I worked on remedying them for a year, and eventually did; sadly, I never got around to rewriting the paper or fixing the errors in complete form. I was a little discouraged by how little interest there was in the hyper operators, so I abandoned it to work on more palpable "mainstream" (if you will) problems.

The final claim of the paper is in fact correct (I can prove it now, but it would take about 30 - 40 pages of reasoning I haven't gotten around to writing out clearly), it is only that I proved somethings within the paper with too much of a hand wave that forego some of the subtleties of the question. In short, I gave an erroneous statement which led to the result. But the erroneous statement can be tweaked slightly to to give a correct statement which still gives the result.

As a professional, I would not cite my paper. It was the first paper I ever wrote and I got ahead of myself very often. I've tried consistently to remove it from arXiv, but that's what arXiv does, it archives. I've been meaning to rewrite the paper, but I haven't found much time for it. Mainly because of how little interest journals expressed towards the problem, and that no one was likely to publish it (disregarding it had a structural error).

All and all, you're in the same boat I was in back when I started writing that paper. There were no real papers to cite. And the papers to cite, were either: obscure, hard to follow, narrow, or never really said what you wanted to be said. Although people will tell you, straightforwardly, and incredibly often "a solution to

exists," but you'll be hard pressed to find three or more papers on the matter written in a professional manner. They were just in the hemisphere of knowledge, but weren't down to earth yet. And as you've complained, these papers tend to not be written fluidly with a broad appeal.

You're in uncharted waters here. This math is fringe math (as I like to call it). Even if there are papers, only a select few know about them. And then, there are a select few who know different papers from other select fews, as though there's a schism between the literature available for the few people working on the same problem. How things are being proved here is similar to how maps of North America were being charted in the late 17th century. They looked pretty close to North America, but weren't actually the truth--and everyone had a different looking map. Consider the qualms of defining the logarithm in Euler's day, no one really knew what was going on when they saw

but it also equals

.

To conclude. Do not cite my paper. If you have questions regarding my paper and qualms or what have you, pm me. I'd be happy to answer any questions. Just, again, my paper isn't worth the paper it's written on (at least mathematically). When one brick is missing at the bottom of a tower, the entire thing collapses. This area of math is largely an oral thing at this point. It's the discussions which are making the field, not the actual written word. To prove this point, Sheldon and I are responsible for a proof that tetration with base

is a completely monotone function. But neither of us bothered to write up a proof. Which is the lay of the land in these parts.

Regards, James.