What Brix is, and why sugar preserves
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What it measures
Soluble solids as a percentage of total weight. In a syrup that is almost entirely sugar, so Brix and sugar concentration are used interchangeably in practice even though they are not the same claim. The instruments read density or refractive index and give you Brix from it, which is why a density figure and a Brix figure describe the same thing.
How sugar preserves
Microorganisms need available water. Dissolved sugar holds water molecules and makes them unavailable, which is measured as water activity rather than as moisture content. Enough sugar and the water in the bottle is there but out of reach, and nothing grows in it.
Why this puts a syrup outside the acidified foods rules
The federal definition of an acidified food applies only to foods with a water activity greater than 0.85. A syrup at a proper density sits below that, so the whole regime that governs a pickle or a sauce does not reach it. That is the single biggest practical difference between this craft and the ones next to it in this library, and it comes from one number.
Which is why the number is not negotiable
A syrup finished a little thin is not a slightly weaker product. It is a product whose water activity may have crossed back over the line that was keeping it safe without refrigeration. The difference between a shelf stable syrup and one that molds is a couple of percentage points, and you cannot see them.
You cannot taste it either
Two syrups a few degrees Brix apart taste similar and pour differently only to somebody who handles a lot of it. There is no sensory test that substitutes for the instrument, which is the whole reason the instrument is the first purchase.