What is the closed form of ridge regression with gaussian family in package "glmnet"

Question

I take the deriavite of the beta in the "glmnet" with gaussian.

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Then get a closed form like this

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When I use this form with the normalized X and normalized Y and get the same solution of the function "glmnet".

But why can't I get a approxmiate solution with the normalized X and center Y.

Here are about the gaussian family in the glmnet vignette.

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