What will be implications of not publishing officially?
It's difficult to answer this question effectively because it is premised on some incorrect assumptions.  Let me begin by correcting them:


There is no such thing as an "official" scientific publication.  There is no Board Of Serious Scientists who determines which publications are legitimate and which are not.  Science is a marketplace of ideas, and many things that have never gone through a typical peer-review process have gained high citations.  For example, the Agile Manifesto has hundreds of citations and vast scientific impact, but has never existed as anything other than a page on various websites.  Thus, you are not "violating" anything, you are just choosing an unusual and often less effective route for getting people to notice the significance of your work.
If you publish work, it doesn't prevent somebody else from publishing the same ideas if they don't know about you.  Parallel invention of concepts happens all of the time in science.  You cannot magically exclude people from working in an area by "getting there first."  If nobody has noticed the significance of your work, it's as though your work doesn't exist.  If they do notice you, then that doesn't stop them either: they'll just build on your work and cite your informal publication.


So, what are the actual advantages and disadvantages of publishing in this informal way?


Advantages: You don't need to convince any other researchers that your work has value, you can just publish it.
Disadvantages: Your work may simply lie unnoticed and disregarded.  Putting the source for an informal publication on GitHub isn't much different than sticking a PDF on your website. 


Let me also answer the question that you haven't asked: how do you get your work to have a scientific impact?  For this, you need to figure out how to engage with the scientific community.  Go to conferences and listen to other people's talks in related areas.  Figure out what questions can be answered by your work that cannot be answered by prior methods.  Above all, learn the language that other researcher use so that you can speak to them in a way that will not set off "crackpot alarms." You may well have created a fabulous new work of mathematical theory, but using words like "entirely new mathematical theory" sets off crackpot alarms, because they are frequently used by crackpots and rarely by researchers who actually have something to contribute.