> It's modular arithmetic because of the internals of the processor, not because of some law-of-the-universeI beg to differ. There were various integer representations tried, such as having a separate "sign" bit on integers. But the two's complement finally was used by everyone. So there's something deeply unique to modulo arithmetics that none of the other representations could offer. This is not just about saving a few gates in the circuit.
> However, it's easiest to just truncate the number and essentially do modular arithmetic
The point is, there is no truncatenation step. It's just a side effect. If you build any naive adder circuit, substractor, multiplicator - even if you totally ignore negative numbers, you automatically have a full working two's complement, that works for negative numbers without any additional effort.
And why is that? Because the mathematical structure behind it is so dead-simple that it almost works by miracle. (except for division, which is highly non-intuitive and has more pitfalls than usual pitfall of division by zero.)
As a consequence of this simple math structure, it is very easy to build in hardware without any additional handling of nasty special cases.