We always have some "map", so technically all we can ever check is whether "map + experimentalData(territory)" is consistent. The trick is to use the map in weird ways, which can exaggerate possible incosistencies.

In given example, let's suppose that the exponent is not 2, but for example 2.001 -- then yes, we would be measuring it using imprecise electronic equipment, but there is a small chance that all those differences would exactly cancel each other out.

As a metaphor, imagine that we have a function "f" that we believe returns f(x)=x^2, but in reality it returns f(x)=x^2.001. Unfortunately, we can never inspect numbers directly, only their f-values. Trivial checks like "f(5) = f(5)?" would not help us discover the problem. Some more complicated checks like "f(2×3) = f(2) × f(3)?" would still give the expected answer. But for example check "f(2+2) = f(2) + f(2) + f(2) + f(2)?" would fail. A complicated test like "f(2×3+2×3) = f(2)×f(3) + f(2)×f(3) + f(2)×f(3) + f(2)×f(3)" will more probably fail that appear correct. -- Using electronic devices seems to me like using these complicated tests; there is very small chance they would fail in exactly the necessary way to make the error in theory invisible.