Mathematics of Cap Charging Circuit
Basically it's a boost converter with no load


Here's your first problem. A theoretical boost converter with no load produces infinite output voltage. Basically it's a power converter - you put energy into the inductor and release that energy into the capacitor. Without a load, the output voltage will continue to rise for each switching cycle.


  I've run a simulation in LTSpice and found that the simulation results
  during the "switch-open phase" do not match my predictions at all


Using a 1N400x diode is a poor choice - the reverse switch off time of circa 30 us means extremely poor performance. Use a diode that can handle fast switching.

Also we do not know what your predictions are.


  So where did I screw up?


Possibly on the two points above.