I (Electrical + Software Engineer) once worked for a physicist who believed that anything less than an order of magnitude was merely an engineering problem. He was usually correct.

I was taught the same. To not care a lot about things under an order of magnitude. Over the years when planning large software projects or assessing incidents and so on, the 1 order of magnitude threshold helped me often.

Bingo, I studied Physics!

As a physicist, I think this is correct too :). You don't start to see problems with things under that, unless they are deviation from standard model predictions.

Variance in accuracy of this statement also safely within one order of magnitude

An order of magnitude isn't a defined quantity, it depends on what base you're working in.

The difference between log 2, ln, and log 10 is less than an order of magnitude, so to a physicist it's all the same :)