Mindless insistence on silence at the day centre (4)

I believe the answer is:

gaga

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'mindless insistence on' is the definition.
I can't tell whether this definition defines the answer.

'silence at the day centre' is the wordplay.
'silence' becomes 'gag' (gagging is a kind of silencing).
'at' means one lot of letters go next to another.
'centre' says to take the centre.
The central letter of 'day' is 'a'.
'gag'+'a'='GAGA'

Can you help me to learn more?

(Other definitions for gaga that I've seen before include "dotty" , "fatuous?" , "Bats" , "Foolishly doting" , "Slangily crazy, off one's head" .)

I've seen this clue in the Evening Standard.

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