The original source of the big bang is not, at this time known.
The current understanding of the unfolding of the universe after the big bang is based on a number of observations that we make AND additional calculations that produce a pretty good description of the current state of matter in the universe.
However, since there are still a number of issues left to resolve there is active, ongoing research in this area. Therefore nothing is "proven" even in the lose colloquial view of the word. Therefore nothing is taken on "faith" even in the rather wide open form of the definitions you have chosen to use. Without explaning why you choose to use them I might add.
The definitions of "faith" that you have used are so different from the formal defintions of "faith" as it applies to what we normally think of as religious matters that it is clear that it doesn't matter if any of this does have some form of "faith" 4 and 5 in it.
None of it has faith type 1 in it. Other than a personal view by many scientists that it will be, in the long term, possible to know how the universe started which maybe be taken as not based on direct evidence.
However, many other areas have been successfully explained by science and moved from the sphere of religions (faith type 2 and 3) to the sphere of science. This could be construed as evidence that this kind of success will be continued. I wouldn't call it very strong evidence but it is not a belief based "not on proof or evidence".
™ Version 4.2