Out of interest, how long do they typically need to stay in place?

I think he meant double the rate of productivity growth, not double productivity. So, from nothing ... to double nothing.

Berlin (same scale as central London)

Central London (smaller scale than previous examples). The answer seems to be 'parks and along the river'.

Brighton. Unsurprisingly, the seafront is popular!

Lewes. A lot of access east/west along the South Downs.

Maybe a 'softer' proof than you intended, but you could set up an economic incentive that is 'proof' that the private key is not currently known. e.g. $100m in a wallet address derived from the key. If you or somebody else had the private key they'd have withdrawn the money.