Understanding of the objective function of the model

[zhanght2120]

Hello,
I'm reaching out for some clarification on the objective function as described in the paper. I've attached an image for reference . Could you explain why the first term in equation (3) is zero?

The authors of the paper state that "the mechanism of 'inference' involves computing the analysis transform of the image and adding uniform noise (as a proxy for quantization)." This process is illustrated in the attached image

However, since the input x is fixed, and I believe that the Encoder outputs y not \tilde{y}, and the output of the Encoder represents the probability distribution of y. Therefore, when we talk about the probability distribution of the Encoder, we should actually refer to q(y), However, since \tilde{y} can be calculated from y, we can deduce q(\tilde{y}).
I've attempted to deduce q(\tilde{y}) in the following manner (Image 3), but I'm unable to see how this leads to the first term in equation (3) being zero. Am I missing something or have I made an error in my reasoning?

I would greatly appreciate your insights into this matter.