What can be done to bridge the gap between amateur and professional mathematicians? I am an amateur mathematician. I have trained myself on math-related websites and other resources. However, the lack of technical knowledge, methods, and sociological culture usually shared by professional mathematicians has prevented me from going further and publishing rigorous and interesting results.

Many amateurs, often outside areas where universities are easy to reach, face the same hurdles. I know a lot of them are cranks but not all of them are: some have extremely good computational skills, an extremely original vision of maths and sometimes a lot of time on their hands.

What can be done to bridge this gap between amateurs and professionals? How can we give amateurs the codes and techniques needed to enrich the corpus of mathematical results with new, fresh and interesting contributions?

For example, could retired professions create a free resource that links universities and labs to amateurs? In this way, the retired professionals could act as "pre-referees," teaching the amateurs how to write a paper, and collaborating on the research itself? There could even be a magazine in which publications resulting from this are published.
First, I can't help but notice your complaints are extremely vague and the language is evasive. You don't like "sociological culture usually shared by professional mathematicians", but it's hard to guess what you mean by that, and what are specific "aforementioned issues" you have problems with. Put it into plain words, please.

However, I can give a general comment. Amateurs might indeed have brilliant ideas, and this is true for all branches of science from history and linguistics to maths and chemistry. Amateurs can write papers and submit it to journals/conferences as we at universities do.

The problem is usually in, as you say, "lack of technical knowledge and methods", which in practice means that these papers are often poorly written (I had to review some) and aren't ready to be published. Your idea of "bridging the gap" by effort of retired professionals is rather popular among amateurs: we might have brilliant ideas, and all we lack is just knowledge of some technicalities, so we just need someone to take care of them, so we can focus on the big things.

Unfortunately, working in science is a full-time job, and dealing with "technicalities" takes most time. I would go as far as saying that understanding the methods and being able to write a decent paper is as important as having bright mind full of cool ideas.

Potential "gap-bridgers" should 1) be good in several wide areas (while academics usually specialize in narrow fields); 2) devote a lot of free time to find grains of brilliance among 99% waste generated by amateurs; 3) do the hardest and least rewarding work by turning mere ideas into something publishable; 4) neglect own interests in favor of pursuing someone else's agenda.

Summing up, an amateur doesn't necessarily have to undergo a formal training and get a PhD. However, having good understanding of methodology and paper writing is vital. There are books devoted to this subject -- take your time and acquire the necessary knowledge rather than waiting for someone to do the job. If you still need help, try to collaborate with people having more experience and write a joint paper.