Doubt that it has anything to do with infusing......
You can't infuse anything into pocket infinity.
And if it were possible, you would have to have a secondary with less than a 170 attack rating in order to infuse pocket infinity into another weapon.
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That's why I'm saying it could be a weapon awarded by the gunsmith that may be able to infuse and unlock the new one. I bet it has a flavor text that says something along the lines of being a key to the past and finding the right doorway.