Going off the comment on the other answer, I believe you are a
U.S. dollar bill.
My products are quite edgy, But most have been a bust,
There have been many types of paper currency, a lot of which have been discontinued.
I'm only one of many,
There are many bills now, both in sheer number and in denominations.
Lord knows who we can trust,
Dollar bills have "In God we trust" printed on them.
I'm not a politician, Yet I work with presidents,
Most dollar bills have presidents on them (they are even sometimes called after the president depicted on them).
And like a mathematician, I seek proofs of excellence,
Not exactly sure here? I think proof bills are printed to check for 'excellence'?
I resemble a detective, Who fills in all the blanks,
Not exactly sure here? I was thinking maybe slang terms, but I'm drawing a blank. (pun intended)
And if gold is your objective, I've more than many banks,
Dollar bills used to be banknotes, which were issued by individual banks rather than the Federal Reserve. Also, they used to be Gold Certificates, where you could trade in the certificate for actual gold with the issuer.