Um... you don’t keep your ketchup in the fridge? Yikes.
I would say that it is the acidity from the tomatoes and vinegar more than the sugar that inhibits bacterial growth. It will not inhibit mold growth. Sugar is after tomato concentrate and vinegar on the ingredient list; ingredients are listed in order of quantity, starting with the highest quantity.
I just looked at pictures of ketchup bottles on line. They all say to refrigerate after opening.
Yes, it’s the acidity that makes it shelf stable. When I waited tables we never refrigerated the ketchup bottles. This is SOP in the restaurant industry. Of course we were cycling through it pretty quickly.
Here’s some commentary including Heinz’s official statement on the matter:
https://www.today.com/food/should-you-keep-ketchup-fridge-or-pantry-t107837
Um... you don’t keep your ketchup in the fridge? Yikes.
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Growing up, my Mom always kept an open bottle of ketchup on a shelf in the kitchen. That along with an open jar of Crisco for baking and frying. I think the Crisco would go bad after many months. The ketchup seemed to be good forever. This all might have been because my Mom grew up in an age when an “ice box” was indeed a box, with ice, so there wasn’t a whole lot of room to store anything inside it.