My impression of local police in China is that they are underlings to the PSB. If your abode is near the PSB, the PSB will check it out. But if it's not, the PSB will demand that the local police check it out

As far as paperwork to prove your accommdation as stated by you and your wife is correct, you will be asked to provide the original of a rental contract., which will have to be photocopied. The landlord's ID card and a photocopy of it will also be necessary. One Chinese marriiage certificate is not enough. They will need to see both, and need photocopies of both of them too.

The PSB won't accept her written statement that she will stay with you until the money runs out. What they need is her signed declaration on the date of your application that she asks them to grant you an RP so that you can stay with her.

Another thng is that visa fees at big PSB premises can only be paid by bank transfer, not in cash.

Finally, don't be caught out telling lies to the PSB. That would invalidate your RP application, MEANING THAT YOU'LL HAVE TO LEAVE CHINA, AND WIFEY WILL BE VERY SAD WITHOUT YOU!
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You put things nicely in perspective for me. At some stage of the application I have been told I will be sent to the big city (small by chinese standards) where secretly we want to live and to move to. Do you see any important complications about that at all, if after receiving our passports back with residence permits we suddenly announce to the local police that we are off to live in the city?

Some poster i think was upset I had made a joke or two, on about me being rustic. But I genuinely appreciate the information and advice you have provided. Many thanks.