Almost interestingly the Royal Mail web site (http://www.royalmail.com/postcode-finder lists the library at SE23 3HZ,
but draws its location further up .... in Sydenham!

I think the RM database (which also supplies the free-postcode-maps website), is batching a bunch of mixed postcodes into rough locations maybe of 100m radius, and the RM website map just points at the centre of this zone for a given address.

I suggest the free-postcode-maps site in turn plots the SE23 boundary around these larger zones and hasn't got a clue where the real boundary is.

My A-Z shows Blythe Hill outside SE23, but it is old. Google maps shows it partially inside and so perhaps FH Soc could indeed approve of tim tower being located there.

Or those that can should dig deeper http://www.guardian.co.uk/artanddesign/2012/nov/09/billionaires-basements-london-houses-architecture