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When adding a constant combinator with a same colored wire to an Arithmetic combinator as another input, the constant input does not show. If I wire it with both colors, it shows and is registered on the one channel, but not the first one.

I have attached 3 screenshots showing of the combinations in a visual manner, which should help decipher the text

Thanks for the report however i am not seeing anything wrong here. If CC sends a signal of gun-turret=50 while the other signal sends a value of gun-turret=-65, then all images look exactly as expected:
1/ CC connected to red: red shows gun-turret=50, green shows gun-turret=-65, as expected. Output is correct (-65 + 50 -> -15)
2/ CC connected to green: green shows gun-turret=-65 + 50 = -15, as expected. Output is correct.
3/ CC connected to both: red shows gun-turret=50 from CC as expected, green shows gun-turret=-65+50 = -15, as expected. Output is correct.