how to do math correct frequency separation of 16 bit RGB?

[kirkr5689]

I copy image as a LF layer and blur it .  then convert the doc to 32 bit    ( if I have any alpha I apply gamma 0.45454 to any alpha channel )      Then I set subtract blending mode to LF layer  and  get looking almost black  HF image  with most pixels negative .    With add blending mode on top of  same blurred original they do  perfectly frequency separated pair. 

But how could I brinng it back to 16 bit integer  mode ?     I need to lift that mostly negative  HiFreq  layer to be around 0,5 right?   So I add 0,5 flat color layer  with add blending  but when I convert the doc back to 16 bit mode the gamma changes  and what was 0,5  is brighter now .  I do gamma 0,454545  to fix it.    Now  I believe   this HiFreq layer should work  on top of blurred original  being blended as LInear Light, right?   But it looks too contrast.   I try hard light  and it also not exactly  right . I am totally puzzled  now.

Could anyone explain me please where my mistake is or perhaps  how could it all be done in 16 bit mode altogether  without converting  into 32 bit mode at all ?   I would be very appreciative  for any suggestion.   Ideally it should work as an action or script.    When chat GPT writes me one  it tends  to use   "Apply Image"  and it is not working for some uncertain reason.  Still I  can't even do it manually with math accurate result .  

PS. I figured how to do it without  32 bit mode conversion.   Just using same subtract idea with 128 offset ( lift) and 2 scale in apply image dialog.   Works just right  in 16 bit integer mode.    Chat GPT tried to lift it  to 32768  for some uncertain reason

[Stephen Marsh · Accepted solution]

@kirkr5689 

For 8 BPC: In Apply Image, use Subtract in the Blending menu, and set Scale to 2 and Offset to 128.

For 16 BPC: In Apply Image, use Add in the Blending menu, check the Invert checkbox and set Scale to 2 and Offset to 0 (zero).

No need for 32 BPC mode.

[kirkr5689]

Thanks a lot  Stephen