I though you are thinking about solving incompressible NS with some kind of projection method. The reason of the time-level difference is that continuous equation should be applied to right hand side term of the equation of pressure. At the next time-step (n+1), the continuous equation is assumed to be satisfied (divU=0), ideally. Residuals are corrected step by step from this assumption. I hope this is useful for you.
I think it is not satisfied mathematically as you thought. When the residual is converged to zero ideally, the equation is satisfied completely. Therefore, it can be satisfied only for the sense of numerically., I think.
Pressure correction equation is a Poison equation, its a time marching problem assuming some pressure as initial condition and march for next time step. In projection scheme, Pressure is used to project the velocity in from non-divergent space to divergence free space, so we need correct pressure at that time-step.
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