I sadly found out that my current result on CI+KF only works in a very limited scenario.
The goal is to prove that CI step can diminish the unobservable space, which means that the information can be meaningfully cascaded by CI step. However, by rigorous defining the algorithm, the information from CI step contains the effect of time update. Therefore, we can not get rid of the nonsingular assumption of $F$, which is the system propagation matrix.
Another insight is that Riccati recursion can be regarded as a general tool for the update in covariance form and information form (the inverse of covariance). And we need a tool to convert between these two description, which relies on the nonsingular assumption of $F$ as well.
To prove the final goal, we have to use Camley-Hamilton theorem as well. I hope I can finish the paper as soon as possible.