Your own bank statement plus a receipt from Bob would be good enough, right?

I'm not sure what "mathematically air-tight" could mean because you can't use math to prove that money changed hands. Any of the data that's going into your math could be faked. Even with current cryptocurrencies, the consensus is not necessarily infallible.

I'm afraid that's the only answer, in that you would need some piece of piece of information back from Bob that Alice is able to corroborate somehow. But then how do you deal with forgeries, the information would have to be corroborated between the two somehow. Perhaps Bob would "publish" who gave him money and how much and perhaps a secret, and then Alice would know the money got to Bob and the secret would be kept. It requires active participation from Bob to publish though, which is not ideal.

There are five entities involved:

1. Alice, who has knowledge of her transfer

2. Me, who has knowledge of both transfers

3. My bank, which has knowledge of both transfers

4. Bob's bank, which has knowledge of both transfers

5. Bob, who has knowledge of my transfer

The question is, whom does Alice trust? If she trusts me, my bank, or Bob's bank, then you don't need any more proof. You just need a statement from any of those entities that documents the transfers.

If she trusts Bob, then her knowledge of her own transfer can be combined with Bob's knowledge of his transfer, and she can put those together.

In real life, the only way to prove anything to Alice without Bob's knowledge of Alice would be to use #3, my bank. This would require that the bank can produce proof that Alice trusts that includes both the sending account (mine) and receiving account (Bob's).