Background Check Failure - Implication on future service

I worked for company A for 10 long years, due to financial issues the company started layoffs - I started looking out for jobs and got myself a good one but with immediate joining. The HR of company A did not agree to relieve me and held by reliving / experience letter. (Though I received the proper documents 3-4 months later with help of my senior, who spoke to HR and got the documents released for me).
I joined company B - the mistake that I commit is, I submitted relieving/experience letter in the company A letter head - signed by self in HR's name. I know this is not at all ethical, but I wanted the job badly and I feared if I tell company B that HR has put my papers on HOLD they may doubt me and my offer will be revoked.
After working for one month in company B, the BGC failed and I was asked to leave. After a gap of 2 months I found job in company C and currently working there, since 2 years. they never asked my about gap nor did i mention
My question is:
1. In new job search/interview, should I mention about this one month old job to prospective employer
2. In case I mention, should I tell him the actual reason for termination/end of service.
3. If I don't tell the prospective employer, will they find out about employment at company B from UAN portal or somewhere else.

Hi,
As you yourself agreed, what you did is not ethical and amounts to forgery. You better be at Company C for some more time and in future when you look for job don't mention about Company B itself. Also don't show it in your Resume.
When you go to new employer you can continue with same UAN of Company C.
Always be ethical in terms of employment and in personal life. Always honesty is the best Policy !

Dear Friend,
If your company A.B,C deducted your PF earlier then your service history through PF portal is accessible to the future company so avoid hiding, let keep honesty with you definitely it worth for you in your future.
If we hide something then the same thing does not allow us to sleep all night.

@V.M.Lakshminarayanan sir
1. If I don't mention company B to next employer - he may get to know about it from UAN
2. If I mention and tell the reason/cause of exit - I stand no chance of employment

Hi,
In what way you got relieved from company B? You have not mentioned about it.
If they have relieved you properly with relieving order then no harm in showing. You may present some fitting reason.

@V.M.Lakshminarayanan sir,
After BGC failure, I was called and asked to leave as per HR policy.
They gave me a relieving letter stating that my service with the company is terminated immediately because of failed BGC due to incorrect information provided by me, and that my dues will be cleared shortly.
They even approved my UAN linkage when I requested them after joining company C.

Hi,
Then better to share with your new employer if query raised by them.

Hi,
you also worked in HR Department, you might know about HR background verification process.
you must be honest to that company & must share what cause of relieving.
P.Packiaraj