Thank you all.
Alan - it is indeed a 5712.
The excitement of unwrapping the watch got the better of me as I elbowed the small adjustment pin which fell on the floor and disappeared (I should have left the pin alone). I just couldn't see it anywhere and assumed it'd disappeared underneath the sideboard
I visited the Patek store, they told me one watch = one pin and that they couldn't get me another. I can somewhat understand they want to avoid giving the pins away!
Luckily my watch dealer promised to source me another. I found it in the end resting in a cavity at the bottom of the sideboard
but not before the motivation from uncooperative attitude at the store had me jacking up the furniture using the 4x4 jack I'd used earlier