A tx now might cost 5k sats, which is approx £2.50.

At £5m per bitcoin, assuming the fee's remain stable, a 5k sat tx now costs the equivalent of £250.

At £50m per bitcoin that tx, again assuming fee's remain stable, £2500.

At that point noone outside of early adopters, business and large institutions are going to make on chain payments.

It's true, sats per vbyte can drop, and they can also rise. UTXO management can also reduce your exposure to higher fee's. But at some price point it doesn't matter what you do, ordinary people will be priced out of on chain transactions.

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if you assume that nobody finds more ways to squeeze more transaction capacity out of the protocol then you are right, but history says otherwise

a lot of bitcoin transaction demand has moved to lightning, more than enough to keep on chain fees competitive with fiat market clearinghouse rates

Fiat market clearing house rates? I think we're talking about different things. If you assume transaction fee's in sats/vbyte are going to drop then maybe...but I doubt that. There'll be periods where it's lower, periods where it's higher, but there'll always be fee's and in £/sat terms, as the bitcoin price rises so does the cost of online transactions, which will price a majority out of on chain use.

yes, competition against fiat money transfer costs is very relevant

for all transactions on chain above about $1000 nominal exchange rate value the fee rates are close to 0.1%, and as you increase your tx size this overhead cost drops proportionally

a large majority of small transactions are now being done off chain with LN... i pay my VPS bill with btcpayserver... many businesses are adopting this easy to deploy system and this is bitcoin payment volume that never sees the chain in any substantial way

Okay we seem to be talking at cross purposes. My maths is right, that much I know absolutely, so I'll happily leave it there 😁