I made a cooked compote with dried fruit and am wondering what safe refrigerated life I can expect. Ideally, please provide some reasonable rationale or reference for the advice.
There are three major things that will effect the "natural" shelf life of the compote, in that they could extend it considerably past the normal shelf life of the ingredients individually:
How sweet is it? If it is sufficiently sugary, to the level of a jam or jelly, the sugar in the compote will act as a preservative.
This is because any bacteria or mold that try to colonize the product will be dessicated, as water exits their cells into the sugar medium via osmosis.
The problem here is that sugar is hydrophylic, and will easily attract water from the environment when the jar is opened, or when a wet spoon is put into it. If the surface becomes diluted with water, and thus the sugar is less concentrated, mold can get a foothold. So this kind of product is best stored in the refrigerator once opened.
How acid is it? Very, very acidic foods are less hospitable to most pathogens.
How salty is it? This one doesn't usually apply to compote recipes, but high enough salt levels also make foods inhospitable to most pathogens, again due to the dessication of their cells via osmotic pressure.
The specific recipe you linked to--at least the compote portion itself, excluding the vinegar syrup and the rest of the recipe--appears to have none of the characteristics that lead to a long shelf life. It should be held no longer than its most vulnerable ingredient, which would be the weak sugar syrup, so it is probably good for several days to a week in the refrigerator.
Now, in general, things named compotes tend to be far sweeter, and far more acidic than the recipe you have linked to.
In these cases, as ElindilTheTall points out, a very, very sweet recipe (jelly-like or jam-like sweetness) will last for many weeks in the refrigerator, and a couple of weeks at least at normal room temperature.
Finally, many compotes are amenable to home canning, if they are sufficiently acidic.
Canning has risks, especially for botulism, so you should only use recipes and methods from a very reputable source when doing canning, to ensure that the product is sufficiently acidic to be safe for the canning method used. Follow all of the techniques and prescriptions in the methods, as well, but I won't turn this into an essay on canning, which is not my area of expertise.
For those compotes that are properly canned, you should get an indefinite shelf life prior to opening, as long as the seal on the canning jar remains intact.
If you placed it in a sterilised jar (boiled for ten minutes, compote added, jar sealed, then boiled for another ten minutes), and assuming it had a decent amount of sugar, it should last weeks even outside of the fridge, especially of unopened. This is standard practice for making homemade jams.
Once opened, it should be kept in the fridge to inhibit mould growth. It should still last a couple of weeks.
If you didn't do all this, I would think it would last a week or two, but there's really too many factors to give an accurate answer. You'll just have to judge its look, smell and taste.
Depending on how the jar was sterilized and the jar quality itself... I'd say indefinite.
I have a few jars of compote in my cellar, made by my grandmother long before my birth. Currently estimating their age at around 50 years. True the natural fruit gelling agents turned the compote into jelly but on special occasions we open a jar of it, and it's always very good, even though you eat it with a spoon instead of drinking.
Note these are original Weck(tm) jars with glass lids and wide rubber seals. The rubber has long turned to dried goo that makes opening the jars a true challenge, and often a hour or more of work, but it seems that only improved the seal. I don't think I had any compote in the modern, metal lid jars to last past 10 years.
The specific recipe you linked to - Transfer compote to impeccably clean containers. Refrigerate for up to 1 week.