A one bit in vector is not assigned when assigment is out of the process

Sure, I know it, but assigment can be also done outside the proces. Note, that line 8 is placed outside the process - it will cuase (in simulation) that bit 0 is not assigned (it get U value)

    CheckOnes_P: process(gray_vector, no_ones_below)
    begin
            for i in 1 to WIDTH_C loop
                no_ones_below(i) <= no_ones_below(i-1) and not gray_vector(i-1);
            end loop;
  ---  no_ones_below(0) <= '1';
    end process;
   no_ones_below(0) <= '1';