Upper Air;
This may be getting pedantic, but when you start quoting the Speed of Sound as 643.3944 kts at standard sea level, something didn't look right.
Standard Sea Level conditions according to the ISA are:-
101.325kPa and 15°C / 288.15°K, and the accepted speed of sound is 661.4788kts / 340.2941m/sec. Just refer to the
ISA Chart in Post #5.
It would appear that a digit or two got transposed in your "magic" multiplier and the standard temperature of 15°C got left behind in the calc.
√ 288.15 *
20.0468023 = 340.2941m/sec
or
√ 288.15 *
38.9678655 = 661.4788kts
So 38.97 seems closer for a memory number.