Lookup dirac delta function and impulse response.
There's no point in discussing this anymore. In your considerable experience maths can often get in the way of a basic understanding of a concept. Everybody else (console/audio manufactors, the science community and dsp developers) simply misunderstand the concept of sign change equals phase shift of 180 degrees of all frequencies because in your experience maths can often get in a way of basic understanding.
Heard of Occam's Razor?
