לע"נ סבי ר' שמואל פנחס בן ר' יעקב צבי

The halacha is the 4 tefachim of סכך פסול serve to disqualify a sukka. What is the din if the סכך has 2 tefachim of סכך פסול next to which is less than three tefachim of air space and then two more tefachim of סכך פסול? In other words, does לבוד serve to disqualify the sukka by connecting the four tefachim of סכך פסול? 

This question was posed by Tosfos [Sukka 17a ד"ה אילו]: 

ומיהו היכא דאיכא שני טפחים סכך פסול ושנים של סכך פסול ואויר מפסיק ביניהם פחות מג', איכא לספוקי אם שני הפסולין מצטרפין לארבעה לפסול את הסוכה דשמא כיון דאויר מפסיק בינתים, לא מצטרפי, או שמא כיון דאין באויר ג', מצטרפין.

Tosfos continues:

ומיהו היכא דליכא בפסולים ד' ואויר פחות מג' מפסיק ביניהם, אין האויר משלימו לארבעה, במה שחושבו כסתום מחמת לבוד...

However, where the two lots of Pasul S'chach combined measure less than four Tefachim, the air in between will not complete the Shi'ur of four Tefachim, due to the fact that it is considered closed on account of L'vud ...

דלא אמרינן לבוד להחמיר

 Because we do not apply לבוד if it involves a stringency.

We have to understand Tosfos. If we don't apply לבוד להחמיר then why can't we resolve Tosfos' first query as to whether we say לבוד to connect the two and two tefachim with the less than three tefachim airspace in between?? The answer should be "NO" because we don't say לבוד להחמיר. We only apply לבוד to be lenient and kasher a sukka. But Tosfos doesn't say that. Why not?

The answer must be that we have to distinguish between לבוד as a "connector" and לבוד as a "closer". לבוד cannot "close" an open area allowing us to see the airspace as sealed up with סכך if that will result in a חומרא [i.e. a disqualification of the sukka]. But we can possibly view the airspace as connecting the two disparate parts [and it is as if there is no airspace] even if that will result in the sukka being פסול. [See the Pri Megadim סי' תרל"ד who talks about the two forms of לבוד]. 

Rebbe Akiva Eiger didn't understand Tosfos' first question, for EITHER WAY the sukka is פסול! If we say לבוד then the airspace connects the two lots of סכך פסול and all together you have 4 tefachim of סכך פסול so the sukka is disqualified. Or even if we see the airspace as being closed with סכך, the sukka will be פסול because the סכך that halachically closes the open area is סכך פסול [because that is the nature of the סכך on both sides]. If we don't apply לבוד the sukka will ALSO be פסול because the small amount of airspace will disqualify the sukka. The only reason we normally require 3 full tefachim of airspace is because less than that qualifies as לבוד. But if we don't apply לבוד here [because it will result in a חומרא] then even less than three tefachim of airspace will disqualify the sukka. So HOWEVER YOU SPIN IT the sukka is pasul so what is Tosfos asking??!!   

The Kehillas Yaakov [סימן י"ג] suggests that indeed we say לבוד meaning that the airspace is viewed as closed off but NOT with סכך פסול as RA"E assumed but as neither סכך פסול nor סכך כשר. Thus there is no airspace but there is also not 4 tefachim of סכך פסול. See his words for more explanation.