Hi Yadgar -

I believe that boxed is a non-repeating pattern, and so is wholly unsuited for
what you want, as well as how you're implementing it.
Read the description - it is ONE box.
You would have to translate it -0.5, rather than -0.1 to even see it.
You have no light source.
You have an odd camera position / look at / with no sky to contrast with.

So in essence you're looking at the unlit black plane that you've defined in
your scene.

Here's my quick dabbling that you can edit and see what I mean:

=======================================================

#version 3.7;
global_settings {
 assumed_gamma 1.0
 ambient_light color rgb <1, 1, 1>
}
include "colors.inc"

// Create an infinite sphere around scene and allow any pigment on it
sky_sphere{ pigment { gradient <0,1,0>
                      color_map { [0.00 rgb <0.6,0.7,1.0>]
                                  [0.35 rgb <0.0,0.1,0.8>]
                                  [0.65 rgb <0.0,0.1,0.8>]
                                  [1.00 rgb <0.6,0.7,1.0>]
                                }
                      scale 2
                    } // end of pigment
          } //end of skysphere -------------------------------------


#declare P_Cubic_Division_Blue_Layer_1 =
pigment
{
   boxed
   color_map
   {
     [0    rgb 0.5]
     [0.5    rgb 0.5]
     [0.5    rgb 1]
     [1    rgb 1]
   }
   translate <0, -0.5, 0>
}

// ACTUAL SCENE

plane
{
   <0, 1, 0>, -1
   texture
   {
     pigment
     { //checker Red, Blue
      P_Cubic_Division_Blue_Layer_1
       scale 100
     }
     finish
     {
       ambient 1
       diffuse 0
     }
   }

}

camera
{
   location <0, 50, -50>
   look_at <0, 20, 100>
   //angle 40
}

light_source {<0, 5, -30> rgb 1}

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