The verse says 'raishis guyim amulaik vi-achreeso aday ovaid' the first of nations is amulaik and it end is certain abolition. We find the abolition of amulaik at the end as Rabbainu teaches that the last letter of 'sheva yepol tzadik vikum' - sevenfold will a tzadik fall and rise, spell out amulaik. The question is where do we see amulaik functioning in the beginning, that the Tora gives him first place.
The Tora teaches that tzoras which spreads to cover the entire body can be considered pure, that is if originally at its inception the tzoras covered the whole body than it is impure, however if it started as a partial nega and afterwards covers the whole entire body that - kulo hufach luvun tuhor - it completey turned white, it is pure.
With this we can understand the function of amulaik at first place. When it is just amulaik completely at the beginning, this is wrotten and impure. However when there are just traces of amulaik at the beginning, but we find amulaik only at the heel (ukaiv, as is sheva yepol tzadik vikum) this is a complete and holy rectification.
This can be understood in many ways. For example it is known that the numerical value - gimatria - of amulaik is the same as 'sufaik', thus we can say, that it is quite proper and in place for there to be uncertainties and doubts at the beginning, so as long as the whole enterprise is not in question. However afterwards at the completion, in order to purify the original sfeykot, there must be complete sufaik. That is complete bitul to G"y, knowing that G"y is Ain Sof without limit in any way, and we really have no idea what we accomplished, and it becomes clear to us that it is very possible that had we taken a completely different course, we may have succeeded infinitely more in sanctifying G"y. However in the beginning we must have complete faith in ourselves, know the sfeykot, and overcome them and persevere. Just afterwards we must achieve the recognition that after all is said and done, we know nothing of what G"y is doing with us.
Mordicay Hatzadik is a model of the above lesson, Chazal teach that the name Mordichay is compramised of the words Mor - dichay, Mor is the name of one of leading fragrances of the Kitores, it is also the gimatria of amulaik. Dichay means pure. This means that Mordichay starts with some aspects of Amulaik, for it starts with the same gimatria, and he ends pure. The final doubt, is found at the end of Mesechet Migela where the Gemara states that all the time Mordichay put into helping save the Jews he wasn't dedicating to studying Tora, and therefore fell in his rank in Sanhedrin. Even though this is a fall which actually earned him greater stength in Torah, an aspect of 'chavolim nuflu lee baneemeen' for if one falls from bina to tiferes, or from hod to yisod, it his an encompassing gain, however the simple intent of the Talmud is not lost, it is clearly stating the aformentioned thought, that at the end of all our deeds we do not know, we simply do not know!
This height - madraiga - of not knowing, was the novelty of Rabbi Nachman who exalted in testifying and even upon occasion swore that at that given time he did not know anything. Also as Rabbi Nachman told Rabbi Nusson that there were times that he, Rabbi Nachman felt such nulification that he was unable to continue without the thought that whatever the case was absolutey nothing forsure it wasn't, for certain he accomplished something in this world.
This is the aspect of the song that is single, double, triple quadruple, Na Nach Nachmu Nachmun MayUman, which as taught in Likutay Meron, is the song of the tzadik which can rectify even the chalal punoy - empty or vacated space that G-d made, hiding Himself so to speak, to create all the worlds. The worlds were created with letters, the chalal is silence, and the Tzadik has the melody which puts everything in its right place and rectification. The first letters of Na Nach Nachmu Nachmun MayUman have the gimatria of amulaik. This is the complete rectification alluded to in the Tora as 'reyshis goyim amulaik'.
Reishis goyim amulaik, the holy books teach, has the acronym Rega - one moment, as it says that G-ds wrath in for one moment, the Zohar explains how this is infact a great attribute of mercy. Upon looking up Tora Reish Eyin Gimel (chapter 273, rega)in Likutay Meron, we find Rabbainu teaching us that the children of the tzadikim, their souls are above the souls of the 600,000 souls of Israel, to such an extant that even when they come into this world and dress themselves in the garments of this world, it is as if they are not here. This means that for these souls they are not blocked by the chalal punoy, they are an aspect of the song of the tzadik.
It now becomes clear how and why it is permissable to enter a bathroom wearing a Na Nach beanie. Because Na Nach is the song of the tzadik which above this world, even when it is in this world, it is not inhibited by the borders and impurities of this world, on the contrary it is a melody which restores the proper place and function of everything in the world.
This is a fullfilment of the fear of Rabbi Chaim Voloziner who states in his sefer Nefesh Hachaim that there will arise chasidim that will study Tora in impure places argueing that the Tora is above the impurities of this world (as in actuality it says in Chazal that the words of Tora can not become impure, however it is obvious and clear that it is forbidden to put them in impure places). The irony is that R. Yehuda HACHASID already taught that there are instances when one can study Tora even in impure places, and that is if a person is overcome with bad thoughts, he is recomended to think thoughts of Tora. This can be explained as follows, there are 50 levels of impurity, after that limit is reached, it can not continue, Kedusha will take place. Also we know, as taught in Lekutay Meron, that even the most impure things have to get their sustenance from the letters of the Tora, the greater the impurity, the greater and more esoteric the powering letters of the Tora will have. Therefore if one is in an impure place and he is overcome with bad thoughts, its as if he's at the last straw, from here on he must reach the place of the highest and greatest realizations of the Tora which can even operate against the worst evil.
So we are thanking G"y for giving us the true tzadik Rabbi Nachman who revealed himself as this holy song, Na Nach Nachmu Nachmun MayUman, which can save us even from the lowest, dirtiest, and impure places there are, and reveal and restore the glorious Kingdom of G-d !