Shulchan Aruch Chapter 12: Things That Invalidate Tzitzit (3 paragraphs)
1. If all the doubled threads on a corner were cut off, but enough remained of all the doubled threads to collectively make a loop, the Tzitzit are Kosher. However, if not enough remained to make a loop, even if only a single doubled string was completely cut off (on both sides) it is invalid. Therefore, given that all threads are folded over into two, if two of the folded over threads were cut off, they are invalid because maybe you cut off the ends of the same thread. Therefore, we have the custom to utilize a sign while making the Tzitzit that differentiates the 4 threads. 4 head-threads are always put on one side of the knot and the corresponding 4 head-threads are on the other side. Therefore, if 4 threads on one side were cut, it is Kosher since all the threads on the other side have more than a loop. However, according to רבינו תם, the Tzitzit are only Kosher if 2 complete long-threads remain, or the corresponding 4 head-threads. The remaining 2 long-threads must be at least 12 thumbs-widths long. According to this view, if 2 different head-threads were cut, if enough remain (of the others) to form a loop, the Tzitzit are Kosher. However, if 3 head-threads were cut off, they are invalid even if a loop could be formed with the remnants. In effect, consider a case where three head-threads were cut off. If you were not (this is an error and so is all italicized words: footnote 5) careful to make the fringes in such a way where 4 head-threads are recognizable on one side of the knot as not belonging to the same mother-thread, we are worried maybe each head-thread belongs a separate long-thread and it would be found that only 1 complete thread remains. Therefore, because of this doubt, it is invalid. But if only 2 head-threads were cut off, it is Kosher as long as enough remain to make a loop. The Halacha is according to the first opinion (not like רבינו תם). However, if at all possible, it is a good idea to heed the opinion of רבינו תם. [The custom is according to רבינו תם. Surely in the case where one was careful to put 4 different head-threads on each side and 3 head-threads were cut off on one side, it is invalid. This is because 3 threads were definitely cut. Similarly, if they were cut on both sides, it is also invalid because maybe they belonged to 3 long-threads.]
2. Consider a case where you need enough thread to tie a loop but due to the strings being too thick, you are unable to form a loop with them but if the strings were thin, there would be long enough to form a loop. In this case, the Tzitzit are Kosher. [One takes into account the thickness of the average thread.]
3. “Enough for a loop”: According to Rashi, this must be done from the “branch”. According to the ר׳י, even if the entire branch was severed but there remains enough thread in the tassel to make a loop, it is Kosher. The world custom is according to Rashi. In a difficult situation, one can rely on the ר׳י.
 Chapter 11 states that in the process of making Tzitzit, 4 threads are folded over making 8 visible threads.
 Lit. “heads” meaning the single threads protruding from the bottom coiled tassels in Tzitzit. I’ll call these “head-threads” to differentiate from the complete threads. The 4 complete threads I’ll refer to as “long-threads” or “doubled over threads” in confusing situations.
 Since only one end of the 4 original strings remain on each side of the knot, if 4 head-threads were cut off on one side, all of the 4 original strings still have enough left over to form a loop because the other side contains 4 long strings; each belonging to one original.
 The head-threads must not belong to the same long-thread.
 M.B: These words are difficult to understand, since if you did not make a sign they are invalid according to all opinions because maybe you cut both ends of the same mother-string. It can be argued that the author meant to place the situation where one did not make a sign in the next case (where 2 head-threads were cut).
Subsequently, the author is referring to a case where you did make a sign and there was a printing mistake. The phrase “if one did not make a sign....” should belong in the next case. However, this is not without difficulty since the Mechaber says “we are worried maybe ... ” and “doubt” while there is no doubt that if 3 strings were cut, they belong to separate long-strings. Perhaps there were 2 printing errors. First, this phrase was misplaced. Then, another printer attempted to correct the error by adding “maybe” and “doubt”. However, this is unlikely since his contemporary, the Rama apparently corrects the Mechaber as well.
 M.B: Following footnote 6, this is referring to a case where one separated the 4 long-threads on each side of a knot. If one did not do this, the Tzitzit are invalid because there is a chance both ends of a double-thread were cut off.
 The “branch” refers to the loose strings hanging out of the tassel.
 Lit. “where it is impossible”