difference between severance period and severance payment period

I was laid off last Oct.  I got one severance document which needed to be signed during 45 days and one severance benefit document.  The severance document clearly stated the termination date and severance payments.  The documentstated after I signed the document, it would send a first payment within a month.  There were total 3 payments in 6 week of period.  After I received the last severance payment, my service would be over 3 years.

The severance benefit document said the service time will be credited during the severance period.  If the service is over 3 years, the pension plan would be vested.  I could get one time lump sum pay from pension plan.

The HR represtative told me that severance period started from termination day.  But the signed severance document has no any word about severance period begin or end definition.  It is very debatable.  My understanding is: the severance period should be the same as severance payment period.

What is your suggestion

Thank you very much for help.