Here is the way I see it, ketchup is probably good for a week or so if you leave it out at room temperature. What really concerns me the most is the surface of the ketchup at the top of the bottle since it was exposed to air after the vaccume seal was broken.
So I store the new bottle out in the open, and then once I break the seal its into the fridge.
But then again, even if I leave it in the fridge for 3-4 weeks I will usually throw it away and open a new fresh bottle.
Resturants do not have this issue because they are constantly using it and refilling bottles