Why the rules exist
Updated
Every requirement on an acidified food traces back to one organism and one number. Knowing why makes the rules easier to follow correctly rather than approximately.
The organism
Clostridium botulinum is a soil bacterium, which means it is on your peppers, your garlic and your carrots as a matter of course. Washing reduces it and does not remove it.
It survives ordinary cooking as a spore. The spore is heat resistant in a way the growing cell is not, so a simmer that kills everything else leaves the spores intact. This is the part that surprises people: cooking is not the control.
Why 4.6
The spore germinates and produces toxin only within a range of conditions. Below about pH 4.6 it does not. That is the whole basis of the number, and it is a biological threshold rather than an administrative one.
The toxin is among the most potent known and is not detectable by taste, smell or appearance. A contaminated jar looks and smells like a good jar. There is no sensory check and no amount of experience substitutes for the measurement.
Why a sealed jar is the risk
The organism is anaerobic: it grows where there is no oxygen. A sealed bottle of sauce at room temperature is close to an ideal environment if the acidity is wrong. Refrigeration slows it; a shelf-stable product has no such help.
So the risk is not in the kitchen, it is on the shelf, weeks later, in someone else's house. That is why the control is a measurement taken before the bottle closes rather than a judgment made when it opens.
What this means at the bench
Target well below 4.6 rather than at it, because ingredients vary and measurement has error. Measure the equilibrated product rather than the mix. Measure in more than one place, because a pocket above the line in a thick sauce is the exact failure mode. And record it, because a batch you cannot evidence is a batch you cannot defend or recall precisely.
None of this is caution for its own sake. It is the shortest path between a recipe and a product that is safe on a shelf for a year.