16 bit file math in PhotoShop

Question

I have been building binary gray scales in Photoshop. The source psd is 16 bit RGB, 1000x1000 pixels. I use the color picker and paint bucket to fill consecutive boxes with ascending binary 8 bit values (color picker only allows selecting 8 bit values). A simple 8 step binary gray scale would be 0, 32, 64, 96, 128, 160, 192, 224, and 255. These values are exactly what I would type into the color picker and confirm on the Info tab.
I then save the file as a 16 bit tiff and read back the RGB values with an external image viewer. In all cases the values in 16 bit are off by multiple 16 bit steps. Instead of 0, 8192,16384, 24576, 32768, 40960, 49432, 57344, & 65535, I read 0, 8224, 16448, 24672, 33153, 41119, 49343, 57567, & 65535. Convert the file to 8 bit and values read 0, 32, 64, 96, 127, 159, 223, and 255.
I do not understand how PhotoShop employs +/- 32768 to define 65536 colors let alone how it converts. My project requires I can predict the values.
Thank you

Answer 1

The conversions are all simple linear scaling with rounding, in integer math.
(32 * 32768 + 127) / 255 = 4112 (in Photoshop 0..32768 16 bit scale)
(4112 * 65535 + 16384) / 32768 = 8224 (scaled for TIFF 0..65535 16 bit scale)
Your expected numbers were incorrect to start with
(32 * 65535 + 127) / 255 = 8224
Looks like you tried to scale by 65536 and 256, and biased towards zero (forgot rounding).
(32 * 65536) / 256 = 8192