> n the "two's complement" representation, there is a sign bit, but the meaning isn't "invert this number", it's "subtract a large, fixed power of 2 from this number".
This is only true if the size of your modulus is fixed. In fact, there is a "sign extension" command allowing you to produce, for example, signed 128-bit values from signed 64-bit values, and this basically requires interpreting two's complement values the same way they represent themselves: a three-bit -1 is just the sum of the positive values +1, +2, and +4. To extend that to six bits, you add the positive values +8, +16, and +32.
The meaning of the high bit in our six-bit scheme shouldn't be viewed as "when this bit is set, subtract 32 from the value instead of adding it". It is "whether this bit is set or not, all higher bits in the number, the ones that don't exist in the hardware, are equal to it"; under this interpretation, the 8-, 16-, and 32- bits were already set in the three-bit value, and that's why they continue to be set when we do a sign extension.
Every place value is still positive, but as long as there is no highest set bit, the number itself will be negative, and equal to the value you'd calculate from its representation as a geometric series.
