The initial thought process might oversimplify the transaction model by focusing on single $B_{scan}$ scenarios. However, a more accurate representation accounts for the complexity introduced when each output is directed to a separate silent payment output, and each signer is responsible for only one input. This adjustment leads to an understanding that the number of proofs required in a transaction is actually a product of the number of recipients and signers involved. Specifically, this is because each signer has the capacity to provide a proof covering multiple inputs, and conversely, each recipient may receive more than one output. This perspective shifts the focus from a linear to a multiplicative relationship, highlighting the intricate interplay between the participants in a transaction and underscoring the importance of considering both dimensions—recipients and signers—to accurately assess the scale and requirements of blockchain transactions.