Schach Pasul Part 2

In a previous post we discussed a problem in the Rambam regarding a board of 4 tefachim wood used in a succah. The Rambam says that although one cannot sleep under the board itself, the rest of the succah remains kosher. The problem is that in general 4 tefachim of pasul schach can invalidate an entire succah.

We explained that a board of wood of 4 tefachim, although not allowed to be used as schach is not a cheftza of schach pasul. This is the nature of the din d'rabbanan. In order to pasul the entire succah, the board of wood needs to have a challos shem of schach pasul.
I must admit, though, that I left out a lot of the background to this issue. The truth is that the mishna itself says that a board of wood 4 tefachim wide does not passul the entire succah. The gemara (Succah 17b) asks on the mishna, why is this so? Doesn't 4 tefachim of schach pasul always passul the whole succah? The gemara answers that the mishna is only speaking when the board is on the side of the succah and we emply dofen akuma to make the rest of the succah kosher.

One may suggest that the Rambam just quoted the mishna verbatim and expected us to fill in the blanks just like the gemara did. Just like the gemara explained the mishna to mean davka when the board of wood is on the side of the succah, so too that's what the Rambam meant all along. This answer is a dochek though, and I would rather try to explain the Rambam according to the pashtus of his words.

So now we are up against a really strong question, because the Rambam is against the gemara. The gemara thinks that a board of 4 tefachim in the middle of the succah should passul a succah while the Rambam holds it doesn't. This seems impossible to get out of.
However, there is hope. The gemara in fact has two separate shittos that hold that 4 tefachim in general can passul the entire succah. There is Rav's opinion and Shmuel's. The gemara only asks on the mishna from Shmuel. They never mention Rav.

This means that in order to answer the Rambam we have to say that the Rambam follows Rav. We also have to identify the difference between why Rav says 4 tefachim of pasul schach passul an entire succah and why Shmuel says the same din. After we identify the lomdishe chilluk between Rav and Shmuel we have to explain why in Rav's approach the Mishna is not difficult and it is only difficult in Shmuel's approach. All of this will have to fit in our original framework that a board of wood for tefachim is not a cheftza of schach pasul. So our mission will be:

1. Find the difference between Rav and Shmuel's seemingly identical statements that 4 tefachim of pasul schach passul the entire succah.

2. Explain why according to Rav a board of wood 4 tefachim won't passul the whole succah while according to Shmuel it will.

I"yh we will pick this up soon.