It's very hard to prove someone "willingly" filed the false report, which in
most states is required to proceed with the charges.   Your state might vary
on this, but I personally know of none that do.

If the complaintent has no evidence to support his accusations the officer
must make a judgment call on whether it's worth it to proceed with what
could be a time (and tax dollars) consuming effort.  Many police reports are
filed for small dollar losses for insurance company purposes.   Very few are
pursued for false filings.

Translation:  "He proved me to be wrong and I'm pissed about it."

You're just pissed because your knowledge of the law is limited to
what you've seen on television or the movies.  If I am "seldom
correct", then I'm sure you can point out a few examples where I was
supposedly wrong.  Go ahead, I'll wait...

Let's see if you can wrap your head around an example.

I go to Sears and buy a set of Craftsman wrenches and put them on my
Sears credit card.  I have the receipt that proves I paid for them.  A
few days later, you do some work on my car and we agree that in
exchange for your labor, I'm going to give you the wrenches.  Nothing
is in writing, because after all, it's just a set of wrenches.

Being a nice set of wrenches, you engrave your driver's license number
on them so they can be uniquely identified if they are lost or stolen.
A couple of years later, you do some more work for me and when you are
done, you accidentally leave the wrenches at my home.  Upon seeing
those nice wrenches again, I decide I want them again, so I keep them
and refuse to give them back.  You go to the police and claim that the
wrenches belong to you, as proved by your DL number on them.  And
since I have them and won't give them back, you consider them stolen
and file a report.

When they question me about the wrenches, I show them the receipt and
the cops tell you since I have the receipt, the wrenches must be mine.
Is that fair?  I have a receipt, but you have a unique number on them
which ties them to you.  Then they tell you since I have "proven" they
are mine, unless you rescind the complaint, you will be charged with
filing a false police report.  Again I ask, would you consider this to
be fair to you?  Now, with all else the same, let's imagine I paid
cash for them, and lost the receipt, but I still claim them as mine.
According to your logic, my claim is all that matters, and even though
they have numbers on them that 3rd party records prove are linked to
you and you alone, you lose your wrenches.  Still want to continue
with this line of argument?

What supporting paperwork?  It's been pointed out to you previously
that the BF said he did NOT have a receipt, and that he likely paid in
cash.  So where is this magical paperwork going to come from?  And how
does this negate the fact he GAVE the phone to her, a fact to which
you've already admitted?  Please tell us, Mr. Lawyer?

I think the more important advice here is never allow a phone you
purchased to be activated on someone else's account unless you are
willing to lose it.  If you want to secure the future ownership of a
cellular phone, in the absence of a written agreement, only activate
it on *your* account.