Again, give an example for the assertions being made.
As for your argument, as others have pointed out, you did not prove anything about the extra length for setting up special output in very specific circumstances, or sketched how that could be accomplished. “Sticking with the most difficult case” you got into the region where you are unable to actually produce an argument. It is far less than obviously true (and may well be false) that the programs which are simple programs but with special output in very specific circumstances, are a notable fraction of large programs.
Again, give an example for the assertions being made.
As for your argument, as others have pointed out, you did not prove anything about the extra length for setting up special output in very specific circumstances, or sketched how that could be accomplished. “Sticking with the most difficult case” you got into the region where you are unable to actually produce an argument. It is far less than obviously true (and may well be false) that the programs which are simple programs but with special output in very specific circumstances, are a notable fraction of large programs.