Warning, the protected name Chi has been redefined and unprotected This is TENverbose, A Maple package, written by Vince Vatter and Doron Zeilberger It is a verbose version of TEN accompanying the article: "A Computer-Generated(!) Proof of the Amazing Loehr-Warrington TEN to the Power n Conjecture" by S. B. Ekhad, V. Vatter, and D. Zeilberger available from the authors' websites. First Written: Aug.-Sept. 2005. Tested for Maple 10. Version of Sept. 21, 2005. The most current version is available on WWW at: http://www.math.rutgers.edu/~zeilberg/tokhniot/TEN . Please report all bugs to: zeilberg at math dot rutgers dot edu . For general help, and a list of the MAIN functions, type "ezra();". For specific help type "ezra(procedure_name);" For a list of the supporting functions type: ezra1(); For specific help for these also type "ezra(procedure_name);" Lemma Number, 1 The vertex, [[-2], [3, -2, 3, -2, -2, 3]], is a clone of vertex, [[], [3, -2]] Proof: The potential mishaps are {[[2, 6], [[3], 3, [-7]]], [[2, 7], [[1], 3, [-5]]], [[2, 4], [[2], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[3], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[1], 3, [-5]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 1 which according to , CrucialLemma[1], are {[[1]]} In the second crack, we have to insert all the good words that sum up to , -5 which according to , CrucialLemma[-5], are {[-2, -2, -2, [1]], [-2, -2, [1], -2], [-2, [1], -2, -2], [[1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[2], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[-2], [3], [3, -2, 3, -2, -2, 3]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 1 Lemma Number, 2 The vertex, [[3, -2, -2, -2, -2], [3, -2, -2, 3]], is a clone of vertex, [[], [3, -2, -2, -2]] Proof: The potential mishaps are {[[1, 6], [[5], -2]], [[6, 8], [[6], 3, [-6]]], [[6, 9], [[4], 3, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[5], -2] It has two parts The List of Fixed Words is, [[], [-2, 3, -2, -2, -2]] In the crack, we have to insert all the good words that sum up to , 5 which according to , CrucialLemma[5], are {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Since this is empty there are no minimal counterexamples coming from this pa\ rticular potential mishap examining the , 2, -th mishap whose potential factorization is , [[6], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 6 which according to , CrucialLemma[6], are {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[4], 3, [-4]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 5], [[2], 3, [-2]]], [[1, 3], [[6], 3, [-6]]], [[1, 4], [[4], 3, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-2]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, -2, -2, 3]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -2 which according to , CrucialLemma[-2], are {[-2], [[1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[6], 3, [-6]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, -2, -2, 3]] In the first crack, we have to insert all the good words that sum up to , 6 which according to , CrucialLemma[6], are {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[4], 3, [-4]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, -2, -2, 3]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 2 Lemma Number, 3 The vertex, [[], [-2, -2, -2, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [w, -2, -2, -2, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [], [-2, -2, -2, 3, -2], equals , -5 The word w must have sum , 5 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[5], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Let's examine them one at a time Enclosing , [-2, 3, -2, 3, 3], between , [[], [-2, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, -2, -2, -2, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, -2, -2, -2, 3, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, 3, -2, [1]], between , [[], [-2, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, 3, [1], -2], between , [[], [-2, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, [1], 3], between , [[], [-2, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, [1], 3, -2, -2, -2, 3, -2] Regarding the meta-word , [3, -2, [1], 3, -2, -2, -2, 3, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, [1]], between , [[], [-2, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3], between , [[], [-2, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, -2, -2, -2, 3, -2] Regarding the meta-word , [-2, [1], 3, 3, -2, -2, -2, 3, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, [1], -2, 3], between , [[], [-2, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, [1], -2, 3, -2, -2, -2, 3, -2] Regarding the meta-word , [3, [1], -2, 3, -2, -2, -2, 3, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3], between , [[], [-2, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, -2, -2, -2, 3, -2] Regarding the meta-word , [[1], -2, 3, 3, -2, -2, -2, 3, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, [1]], between , [[], [-2, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, [1], -2, 3, [1], -2, -2, -2, 3, -2] Regarding the meta-word , [-2, [1], 3, [1], -2, 3, [1], -2, -2, -2, 3, -2], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, [1]], between , [[], [-2, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, 3, -2] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 3 Lemma Number, 4 The vertex, [[3, -2, -2, 3], [3, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, 3, w, 3, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, 3], [3, -2, 3], equals , 6 The word w must have sum , -6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, [1], -2, -2, [1]], between , [[3, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, [1], -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, -2, -2, 3, -2, -2, [1], -2, -2, [1], 3, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, [1], -2, 3, -2, 3] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, [1], -2, 3, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, [1], -2, 3, -2, 3] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, [1], -2, 3, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2], between , [[3, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, 3, -2, 3] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2], between , [[3, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, -2, 3, -2, 3] Regarding the meta-word , [3, -2, -2, 3, -2, -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 4 Lemma Number, 5 The vertex, [[-2, -2, 3, -2, -2, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, -2, 3, -2, -2, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, -2, 3, -2, -2, 3], [3, 3], equals , 4 The word w must have sum , -4 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-4], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} Let's examine them one at a time Enclosing , [-2, -2], between , [[-2, -2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, -2, 3, -2, -2, 3, 3] Regarding the meta-word , [-2, -2, 3, -2, -2, 3, -2, -2, 3, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1]], between , [[-2, -2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, -2, 3, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [-2, -2, 3, -2, -2, 3, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1]], between , [[-2, -2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, -2, 3, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, -2, 3, -2, -2, 3, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2], between , [[-2, -2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, -2, 3, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, -2, 3, -2, -2, 3, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 5 Lemma Number, 6 The vertex, [[3, -2, -2, -2, -2], [3, 3]], is a clone of vertex, [[], [3, -2]] Proof: The potential mishaps are {[[1, 6], [[5], -2, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[5], -2, [-4]] It has three parts The List of Fixed Words is, [[], [-2], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 5 which according to , CrucialLemma[5], are {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[2], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, 3]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 6 Lemma Number, 7 The vertex, [[3, 3], [3, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, 3, w, 3, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, 3], [3, 3, -2], equals , 10 The word w must have sum , -10 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-10], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [-2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1], -2], [-2, -2, [1], -2, -2, [1], -2, -2], [-2, [1], -2, -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, -2, [1], -2], [-2, [1], -2, -2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2], [[1], -2, -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1], -2, -2], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2], [-2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, -2, 3, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1], -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, 3, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, 3, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, 3, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, -2, -2, 3, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1], -2], [-2, -2, -2, -2, 3, -2, -2, [1], -2], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[-2, -2, -2, 3, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1], -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's examine them one by one Let's consider , [-2, -2, -2, 3, -2, -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, 3, -2]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, -2, 3, -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, 3, -2]], getting the word [3, 3, -2, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, 3, -2, -2, -2, [1], -2] . and enclose it inside , [[3, 3], [3, 3, -2]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, 3, -2]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2] . and enclose it inside , [[3, 3], [3, 3, -2]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], -2], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1], -2], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 7 Lemma Number, 8 The vertex, [[-2, -2, 3, -2, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, -2, 3, -2, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, -2, 3, -2, 3], [3, 3], equals , 6 The word w must have sum , -6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, [1], -2, -2, [1]], between , [[-2, -2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1]], between , [[-2, -2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2], between , [[-2, -2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1]], between , [[-2, -2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2], between , [[-2, -2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2], between , [[-2, -2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2], between , [[-2, -2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, -2, -2, -2, 3, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, -2, -2, -2, 3, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 8 Lemma Number, 9 The vertex, [[3, -2, -2, -2, -2], [3, 3, -2]], is a clone of vertex, [[], [3, -2, -2]] Proof: The potential mishaps are {[[1, 6], [[5], -2, [-2]]], [[6, 9], [[7], 3, [-9]]]} examining the , 1, -th mishap whose potential factorization is , [[5], -2, [-2]] It has three parts The List of Fixed Words is, [[], [-2], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 5 which according to , CrucialLemma[5], are {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -2 which according to , CrucialLemma[-2], are {[-2], [[1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[7], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 7 which according to , CrucialLemma[7], are {[3, 3, [1]], [3, [1], -2, 3, 3, -2, [1]], [3, [1], -2, 3, 3, [1], -2], [3, -2, [1], 3, 3, -2, [1]], [3, -2, [1], 3, 3, [1], -2], [3, -2, [1], 3, [1], -2, 3], [3, -2, 3, -2, 3, 3, -2, [1]], [3, -2, 3, -2, 3, 3, [1], -2], [3, -2, 3, 3], [-2, 3, 3, 3], [-2, 3, 3, -2, 3, -2, 3, [1]], [-2, [1], 3, 3, 3, -2, [1]], [-2, [1], 3, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1], 3], [-2, [1], 3, 3, -2, 3, [1]], [-2, [1], 3, 3, [1], -2, 3], [-2, [1], 3, [1], -2, 3, 3], [[1], -2, 3, 3, 3, -2, [1]], [[1], -2, 3, 3, 3, [1], -2], [[1], -2, 3, 3, -2, [1], 3], [[1], -2, 3, 3, -2, 3, [1]], [[1], -2, 3, 3, [1], -2, 3], [-2, 3, 3, [1], -2, 3, [1]], [-2, 3, -2, 3, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, 3, [1], -2], [-2, 3, -2, 3, 3, -2, [1], 3], [-2, 3, -2, 3, 3, -2, 3, [1]], [-2, 3, -2, 3, 3, [1], -2, 3], [-2, 3, 3, -2, [1], 3, [1]], [-2, 3, 3, -2, 3, 3, -2, [1]], [-2, 3, 3, -2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[4], 3, [-6]]], [[1, 4], [[2], 3, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, 3, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[2], 3, [-4]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, 3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 9 Lemma Number, 10 The vertex, [[-2, -2, 3, -2, -2, -2, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, -2, 3, -2, -2, -2, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, -2, 3, -2, -2, -2, 3], [3, 3], equals , 2 The word w must have sum , -2 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-2], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2], [[1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2], between , [[-2, -2, 3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, -2, -2, 3, -2, 3, 3] Regarding the meta-word , [-2, -2, 3, -2, -2, -2, 3, -2, 3, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1]], between , [[-2, -2, 3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, -2, -2, 3, [1], -2, -2, [1], 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[[1], -2, -2, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[[1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1]], [-2, 3, -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1]] } Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1]]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1]] . and enclose it inside , [[-2, -2, 3, -2, -2, -2, 3], [3, 3]], getting the word [-2, -2, 3, -2, -2, -2, 3, 3, -2, -2, -2, [1], 3, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, -2, 3], [3, -2, -2, -2, 3, -2], [3, -2, -2, -2, -2, [1], 3, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, -2, 3]} Let's examine them one by one Let's consider , [3, -2, -2, -2, -2, 3] . and enclose it inside , [[-2, -2, 3, -2, -2, -2, 3], [3, 3]], getting the word [-2, -2, 3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, 3, 3] Regarding the meta-word , [-2, -2, 3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, 3, 3], we have that the factor from position, 7, to position , 12, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 10 Lemma Number, 11 The vertex, [[3, -2, -2, 3], [3, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, 3, w, 3, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, 3], [3, 3, -2], equals , 6 The word w must have sum , -6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, [1], -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, -2, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2], between , [[3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3, -2] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2], between , [[3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, -2, 3, 3, -2] Regarding the meta-word , [3, -2, -2, 3, -2, -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 11 Lemma Number, 12 The vertex, [[-2, -2], [3, 3, -2]], is a clone of vertex, [[], []] Proof: The potential mishaps are {[[3, 6], [[6], 3, [-9]]]} examining the , 1, -th mishap whose potential factorization is , [[6], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], []] In the first crack, we have to insert all the good words that sum up to , 6 which according to , CrucialLemma[6], are {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 12 Lemma Number, 13 The vertex, [[-2], [3, -2, 3, -2, 3, -2]], is a clone of vertex, [[], [3, -2]] Proof: The potential mishaps are {[[2, 6], [[3], 3, [-7]]], [[2, 8], [[4], 3, [-8]]], [[2, 4], [[2], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[3], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[4], 3, [-8]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -8 which according to , CrucialLemma[-8], are {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[2], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[-2], [3], [3, -2, 3, -2, 3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 13 Lemma Number, 14 The vertex, [[-2, 3, -2, -2, 3], [3, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, -2, -2, 3, w, 3, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, -2, -2, 3], [3, 3, -2], equals , 4 The word w must have sum , -4 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-4], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} Let's examine them one at a time Enclosing , [-2, -2], between , [[-2, 3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, -2, 3, 3, -2] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, -2, 3, 3, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1]], between , [[-2, 3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, [1], -2, -2, [1], 3, 3, -2] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, [1], -2, -2, [1], 3, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1]], between , [[-2, 3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, [1], -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [-2, 3, -2, -2, 3, [1], -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2], between , [[-2, 3, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, [1], -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [-2, 3, -2, -2, 3, [1], -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 14 Lemma Number, 15 The vertex, [[3, -2, 3], [3, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, 3, w, 3, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, 3], [3, -2, 3], equals , 8 The word w must have sum , -8 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-8], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, -2, 3, -2, 3] Regarding the meta-word , [3, -2, 3, -2, -2, -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1]], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, [1], -2, 3, -2, 3] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, [1], -2, 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, [1], -2, 3, -2, 3] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, [1], -2, 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, 3, -2, 3] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, [1], -2, 3, -2, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, [1], -2, 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, 3, -2, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2], between , [[3, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 15 Lemma Number, 16 The vertex, [[3, -2, -2, -2, -2], [3, 3, -2, 3, -2]], is a clone of vertex, [[], []] Proof: The potential mishaps are {[[6, 9], [[6], 3, [-9]]], [[6, 11], [[7], 3, [-10]]], [[1, 6], [[5], -2, [-3]]]} examining the , 1, -th mishap whose potential factorization is , [[6], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], []] In the first crack, we have to insert all the good words that sum up to , 6 which according to , CrucialLemma[6], are {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[7], 3, [-10]] It has three parts The List of Fixed Words is, [[], [3], []] In the first crack, we have to insert all the good words that sum up to , 7 which according to , CrucialLemma[7], are {[3, 3, [1]], [3, [1], -2, 3, 3, -2, [1]], [3, [1], -2, 3, 3, [1], -2], [3, -2, [1], 3, 3, -2, [1]], [3, -2, [1], 3, 3, [1], -2], [3, -2, [1], 3, [1], -2, 3], [3, -2, 3, -2, 3, 3, -2, [1]], [3, -2, 3, -2, 3, 3, [1], -2], [3, -2, 3, 3], [-2, 3, 3, 3], [-2, 3, 3, -2, 3, -2, 3, [1]], [-2, [1], 3, 3, 3, -2, [1]], [-2, [1], 3, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1], 3], [-2, [1], 3, 3, -2, 3, [1]], [-2, [1], 3, 3, [1], -2, 3], [-2, [1], 3, [1], -2, 3, 3], [[1], -2, 3, 3, 3, -2, [1]], [[1], -2, 3, 3, 3, [1], -2], [[1], -2, 3, 3, -2, [1], 3], [[1], -2, 3, 3, -2, 3, [1]], [[1], -2, 3, 3, [1], -2, 3], [-2, 3, 3, [1], -2, 3, [1]], [-2, 3, -2, 3, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, 3, [1], -2], [-2, 3, -2, 3, 3, -2, [1], 3], [-2, 3, -2, 3, 3, -2, 3, [1]], [-2, 3, -2, 3, 3, [1], -2, 3], [-2, 3, 3, -2, [1], 3, [1]], [-2, 3, 3, -2, 3, 3, -2, [1]], [-2, 3, 3, -2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -10 which according to , CrucialLemma[-10], are {[-2, -2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [-2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1], -2], [-2, -2, [1], -2, -2, [1], -2, -2], [-2, [1], -2, -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, -2, [1], -2], [-2, [1], -2, -2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2], [[1], -2, -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1], -2, -2], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2], [-2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[5], -2, [-3]] It has three parts The List of Fixed Words is, [[], [-2], []] In the first crack, we have to insert all the good words that sum up to , 5 which according to , CrucialLemma[5], are {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -3 which according to , CrucialLemma[-3], are {[-2, -2, [1]], [-2, [1], -2], [[1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 16 Lemma Number, 17 The vertex, [[3, -2, -2, -2, 3], [3, 3, -2, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, -2, 3, w, 3, 3, -2, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, -2, 3], [3, 3, -2, 3, -2], equals , 5 The word w must have sum , -5 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-5], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, [1]], [-2, -2, [1], -2], [-2, [1], -2, -2], [[1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2, -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2], between , [[3, -2, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, -2, 3, -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2], between , [[3, -2, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, [1], -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, -2, 3, -2, [1], -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2], between , [[3, -2, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, -2, 3, [1], -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[[1], -2, -2, [1], -2, -2, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[[1], -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1], -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1], -2, -2, [1]]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1], -2, -2, [1]] . and enclose it inside , [[3, -2, -2, -2, 3], [3, 3, -2, 3, -2]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 17 Lemma Number, 18 The vertex, [[-2, 3], [-2, 3, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, w, -2, 3, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3], [-2, 3, -2, 3], equals , 3 The word w must have sum , -3 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-3], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, [1]], [-2, [1], -2], [[1], -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, [1]], between , [[-2, 3], [-2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, [1], -2, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, [1], -2, 3, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2], between , [[-2, 3], [-2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, [1], -2, -2, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, [1], -2, -2, 3, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2], between , [[-2, 3], [-2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, 3, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 18 Lemma Number, 19 The vertex, [[-2], [3, -2, -2, 3, -2, 3, -2]], is a clone of vertex, [[], [3, -2, -2]] Proof: The potential mishaps are {[[2, 4], [[4], 3, [-6]]], [[2, 5], [[2], 3, [-4]]], [[2, 7], [[3], 3, [-5]]], [[2, 9], [[4], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[2], 3, [-4]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[3], 3, [-5]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -5 which according to , CrucialLemma[-5], are {[-2, -2, -2, [1]], [-2, -2, [1], -2], [-2, [1], -2, -2], [[1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 4, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[4], 3, [-6]]], [[1, 4], [[2], 3, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[-2], [3], [3, -2, -2, 3, -2, 3, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[2], 3, [-4]] It has three parts The List of Fixed Words is, [[-2], [3], [3, -2, -2, 3, -2, 3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 19 Lemma Number, 20 The vertex, [[-2], [-2, -2, -2, 3, -2, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, w, -2, -2, -2, 3, -2, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2], [-2, -2, -2, 3, -2, 3, -2], equals , -6 The word w must have sum , 6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} Let's examine them one at a time Enclosing , [3, 3], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, [1], -2, 3, [1]], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, [1], -2, 3, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, 3, [1], -2, 3, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, 3, -2, [1]], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, -2, 3, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, 3, [1], -2], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, -2, 3, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, [1], 3], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, [1], 3, -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, -2, 3, 3, -2, [1], 3, -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 7, to position , 12, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, 3, [1]], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, -2, 3, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, [1], -2, 3], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, [1], -2, 3, -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, -2, 3, 3, [1], -2, 3, -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 7, to position , 12, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, [1]], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, 3, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, -2, 3, -2, 3, 3, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, [1], 3, [1]], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, [1], 3, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, 3, -2, [1], 3, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, 3, -2, [1]], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, 3, [1], -2], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, [1]], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, [1], 3, 3, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, -2, [1], 3, 3, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, 3, -2, [1]], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, [1], 3, [1], -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, -2, [1], 3, [1], -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 8, to position , 12, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, 3, [1], -2], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, [1], 3, [1], -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, -2, [1], 3, [1], -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 8, to position , 12, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, -2, 3, [1]], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, [1]], between , [[-2], [-2, -2, -2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], -2, 3, 3, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [-2, [1], -2, 3, 3, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 20 Lemma Number, 21 The vertex, [[-2, 3, 3], [3, 3, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, 3, w, 3, 3, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, 3], [3, 3, -2, 3], equals , 11 The word w must have sum , -11 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-11], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, -2, [1], -2], [-2, -2, -2, -2, [1], -2, -2], [-2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, -2, [1], -2, -2, -2, -2], [-2, [1], -2, -2, -2, -2, -2], [-2, [1], -2, -2, -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2], [[1], -2, -2, -2, -2, -2, [1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, [1], -2, -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, -2, [1], -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, -2, [1], -2, -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, [1], -2, -2, -2, 3, 3, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, [1], -2, -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, [1], -2, -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3 ] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, -2, -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[-2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[-2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, -2, -2, -2, 3, -2, -2, -2], [-2, -2, -2, 3, -2, -2, -2, -2], [-2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[-2, -2, -2, -2, 3, -2, -2, -2], [-2, -2, -2, 3, -2, -2, -2, -2], [-2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's examine them one by one Let's consider , [-2, -2, -2, -2, 3, -2, -2, -2] . and enclose it inside , [[-2, 3, 3], [3, 3, -2, 3]], getting the word [-2, 3, 3, -2, -2, -2, -2, 3, -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, -2, 3, -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, 3, -2, -2, -2, -2] . and enclose it inside , [[-2, 3, 3], [3, 3, -2, 3]], getting the word [-2, 3, 3, -2, -2, -2, 3, -2, -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, 3, -2, -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1]] . and enclose it inside , [[-2, 3, 3], [3, 3, -2, 3]], getting the word [-2, 3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]] . and enclose it inside , [[-2, 3, 3], [3, 3, -2, 3]], getting the word [-2, 3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2] . and enclose it inside , [[-2, 3, 3], [3, 3, -2, 3]], getting the word [-2, 3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]] , [3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, -2], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, -2], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 21 Lemma Number, 22 The vertex, [[3, 3], [3, -2, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, 3, w, 3, -2, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, 3], [3, -2, -2, 3], equals , 8 The word w must have sum , -8 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-8], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [3, 3, -2, -2, -2, -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, [1], 3, -2, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, [1], 3, -2, -2, 3] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, [1], 3, -2, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, [1], -2, 3, -2, -2, 3] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, [1], -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1]], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, [1], 3, -2, -2, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, [1], 3, -2, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, [1], -2, 3, -2, -2, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, [1], -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, 3, -2, -2, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, [1], 3, -2, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, [1], 3, -2, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, [1], -2, 3, -2, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, [1], -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, 3, -2, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, -2, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[-2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[-2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, -2, -2, 3, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[-2, -2, -2, 3, -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's examine them one by one Let's consider , [-2, -2, -2, 3, -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, -2, -2, 3]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], 3, -2, -2, 3] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], 3, -2, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]] . and enclose it inside , [[3, 3], [3, -2, -2, 3]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, -2, -2, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 22 Lemma Number, 23 The vertex, [[], [-2, 3, -2, -2, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [w, -2, 3, -2, -2, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [], [-2, 3, -2, -2, -2], equals , -5 The word w must have sum , 5 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[5], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Let's examine them one at a time Enclosing , [-2, 3, -2, 3, 3], between , [[], [-2, 3, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, -2, 3, -2, -2, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, -2, 3, -2, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, 3, -2, [1]], between , [[], [-2, 3, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, 3, -2, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [3, 3, [1], -2], between , [[], [-2, 3, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, 3, -2, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [3, -2, [1], 3], between , [[], [-2, 3, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, [1], 3, -2, 3, -2, -2, -2] Regarding the meta-word , [3, -2, [1], 3, -2, 3, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, [1]], between , [[], [-2, 3, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, 3, -2, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, [1], 3, 3], between , [[], [-2, 3, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, -2, 3, -2, -2, -2] Regarding the meta-word , [-2, [1], 3, 3, -2, 3, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, [1], -2, 3], between , [[], [-2, 3, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, [1], -2, 3, -2, 3, -2, -2, -2] Regarding the meta-word , [3, [1], -2, 3, -2, 3, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3], between , [[], [-2, 3, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, -2, 3, -2, -2, -2] Regarding the meta-word , [[1], -2, 3, 3, -2, 3, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, [1]], between , [[], [-2, 3, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, [1], -2, 3, [1], -2, 3, -2, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, 3, 3, [1]], between , [[], [-2, 3, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, 3, -2, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, 3, [1]], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, 3, [1]], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, 3, -2, -2, [1], 3, [1], -2], [3, -2, 3, -2, 3], [-2, 3, 3, -2, 3], [3, 3, -2, -2, 3], [3, -2, 3, 3, -2], [-2, 3, 3, 3, -2], [3, 3, -2, 3, -2], [3, 3, 3, -2, -2], [3, 3, -2, [1], 3, [1], -2, -2], [3, -2, 3, -2, [1], 3, [1], -2], [-2, 3, -2, 3, [1], -2, 3, [1]], [-2, -2, 3, 3, [1], -2, 3, [1]], [-2, -2, [1], 3, [1], -2, 3, [1], -2, 3, [1]], [-2, 3, 3, -2, [1], 3, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, 3, -2, 3], [-2, 3, 3, -2, 3], [3, 3, -2, -2, 3], [3, -2, 3, 3, -2], [-2, 3, 3, 3, -2], [3, 3, -2, 3, -2], [3, 3, 3, -2, -2], [-2, -2, 3, 3, [1], -2, 3, [1]], [-2, 3, 3, -2, [1], 3, [1], -2]} Let's examine them one by one Let's consider , [3, -2, 3, -2, 3] . and enclose it inside , [[], [-2, 3, -2, -2, -2]], getting the word [3, -2, 3, -2, 3, -2, 3, -2, -2, -2] Regarding the meta-word , [3, -2, 3, -2, 3, -2, 3, -2, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, 3, 3, -2, 3] . and enclose it inside , [[], [-2, 3, -2, -2, -2]], getting the word [-2, 3, 3, -2, 3, -2, 3, -2, -2, -2] Regarding the meta-word , [-2, 3, 3, -2, 3, -2, 3, -2, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, 3, -2, -2, 3] . and enclose it inside , [[], [-2, 3, -2, -2, -2]], getting the word [3, 3, -2, -2, 3, -2, 3, -2, -2, -2] Regarding the meta-word , [3, 3, -2, -2, 3, -2, 3, -2, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, 3, 3, -2] . and enclose it inside , [[], [-2, 3, -2, -2, -2]], getting the word [3, -2, 3, 3, -2, -2, 3, -2, -2, -2] Regarding the meta-word , [3, -2, 3, 3, -2, -2, 3, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, 3, 3, 3, -2] . and enclose it inside , [[], [-2, 3, -2, -2, -2]], getting the word [-2, 3, 3, 3, -2, -2, 3, -2, -2, -2] Regarding the meta-word , [-2, 3, 3, 3, -2, -2, 3, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, 3, -2, 3, -2] . and enclose it inside , [[], [-2, 3, -2, -2, -2]], getting the word [3, 3, -2, 3, -2, -2, 3, -2, -2, -2] Regarding the meta-word , [3, 3, -2, 3, -2, -2, 3, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, 3, 3, -2, -2] . and enclose it inside , [[], [-2, 3, -2, -2, -2]], getting the word [3, 3, 3, -2, -2, -2, 3, -2, -2, -2] Regarding the meta-word , [3, 3, 3, -2, -2, -2, 3, -2, -2, -2], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, 3, 3, [1], -2, 3, [1]] . and enclose it inside , [[], [-2, 3, -2, -2, -2]], getting the word [-2, -2, 3, 3, [1], -2, 3, [1], -2, 3, -2, -2, -2] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [-2, 3, 3, -2, [1], 3, [1], -2] . and enclose it inside , [[], [-2, 3, -2, -2, -2]], getting the word [-2, 3, 3, -2, [1], 3, [1], -2, -2, 3, -2, -2, -2] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[-2, -2, 3, 3, [1], -2, 3, [1]], [-2, 3, 3, -2, [1], 3, [1], -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[-2, -2, 3, 3, [1], -2, 3, [1]], [-2, 3, 3, -2, [1], 3, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, -2, 3, 3, 3, -2, -2, 3, [1]], [-2, -2, 3, 3, -2, 3, -2, 3, [1]], [-2, -2, 3, 3, -2, [1], 3, [1], -2, -2, 3, [1]], [-2, 3, 3, -2, 3, -2, 3, [1], -2], [-2, 3, 3, -2, -2, 3, 3, [1], -2], [-2, 3, 3, -2, -2, [1], 3, [1], -2, 3, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 23 Lemma Number, 24 The vertex, [[], [-2, -2, 3, -2, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [w, -2, -2, 3, -2, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [], [-2, -2, 3, -2, -2], equals , -5 The word w must have sum , 5 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[5], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Let's examine them one at a time Enclosing , [-2, 3, -2, 3, 3], between , [[], [-2, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, -2, -2, 3, -2, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, -2, -2, 3, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, 3, -2, [1]], between , [[], [-2, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, 3, -2, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, 3, -2, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, 3, [1], -2], between , [[], [-2, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, 3, -2, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, 3, -2, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, [1], 3], between , [[], [-2, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, [1], 3, -2, -2, 3, -2, -2] Regarding the meta-word , [3, -2, [1], 3, -2, -2, 3, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, [1]], between , [[], [-2, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, 3, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, [1], 3, 3], between , [[], [-2, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, -2, -2, 3, -2, -2] Regarding the meta-word , [-2, [1], 3, 3, -2, -2, 3, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, [1], -2, 3], between , [[], [-2, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, [1], -2, 3, -2, -2, 3, -2, -2] Regarding the meta-word , [3, [1], -2, 3, -2, -2, 3, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3], between , [[], [-2, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, -2, -2, 3, -2, -2] Regarding the meta-word , [[1], -2, 3, 3, -2, -2, 3, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, [1]], between , [[], [-2, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, [1], -2, 3, [1], -2, -2, 3, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, 3, 3, [1]], between , [[], [-2, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, 3, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[3, -2, 3, [1]], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[3, -2, 3, [1]], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, 3, -2, 3], [-2, 3, 3, -2, 3], [3, -2, 3, 3, -2], [-2, 3, 3, 3, -2], [3, -2, 3, -2, [1], 3, [1], -2], [-2, 3, -2, 3, [1], -2, 3, [1]], [-2, -2, 3, 3, [1], -2, 3, [1]], [-2, -2, [1], 3, [1], -2, 3, [1], -2, 3, [1]], [-2, 3, 3, -2, [1], 3, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, 3, -2, 3], [-2, 3, 3, -2, 3], [3, -2, 3, 3, -2], [-2, 3, 3, 3, -2], [-2, -2, 3, 3, [1], -2, 3, [1]], [-2, 3, 3, -2, [1], 3, [1], -2]} Let's examine them one by one Let's consider , [3, -2, 3, -2, 3] . and enclose it inside , [[], [-2, -2, 3, -2, -2]], getting the word [3, -2, 3, -2, 3, -2, -2, 3, -2, -2] Regarding the meta-word , [3, -2, 3, -2, 3, -2, -2, 3, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, 3, 3, -2, 3] . and enclose it inside , [[], [-2, -2, 3, -2, -2]], getting the word [-2, 3, 3, -2, 3, -2, -2, 3, -2, -2] Regarding the meta-word , [-2, 3, 3, -2, 3, -2, -2, 3, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, 3, 3, -2] . and enclose it inside , [[], [-2, -2, 3, -2, -2]], getting the word [3, -2, 3, 3, -2, -2, -2, 3, -2, -2] Regarding the meta-word , [3, -2, 3, 3, -2, -2, -2, 3, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, 3, 3, 3, -2] . and enclose it inside , [[], [-2, -2, 3, -2, -2]], getting the word [-2, 3, 3, 3, -2, -2, -2, 3, -2, -2] Regarding the meta-word , [-2, 3, 3, 3, -2, -2, -2, 3, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, 3, 3, [1], -2, 3, [1]] . and enclose it inside , [[], [-2, -2, 3, -2, -2]], getting the word [-2, -2, 3, 3, [1], -2, 3, [1], -2, -2, 3, -2, -2] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [-2, 3, 3, -2, [1], 3, [1], -2] . and enclose it inside , [[], [-2, -2, 3, -2, -2]], getting the word [-2, 3, 3, -2, [1], 3, [1], -2, -2, -2, 3, -2, -2] Regarding the meta-word , [-2, 3, 3, -2, [1], 3, [1], -2, -2, -2, 3, -2, -2], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {[-2, -2, 3, 3, [1], -2, 3, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[-2, -2, 3, 3, [1], -2, 3, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, -2, 3, 3, 3, -2, -2, 3, [1]], [-2, -2, 3, 3, -2, 3, -2, 3, [1]], [-2, -2, 3, 3, -2, [1], 3, [1], -2, -2, 3, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 24 Lemma Number, 25 The vertex, [[], [-2, -2, -2, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [w, -2, -2, -2, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [], [-2, -2, -2, -2, 3], equals , -5 The word w must have sum , 5 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[5], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Let's examine them one at a time Enclosing , [-2, 3, -2, 3, 3], between , [[], [-2, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, -2, -2, -2, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, 3, -2, -2, -2, -2, 3], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, 3, -2, [1]], between , [[], [-2, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, 3, [1], -2], between , [[], [-2, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, [1], 3], between , [[], [-2, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, [1], 3, -2, -2, -2, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [3, -2, 3, [1]], between , [[], [-2, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3], between , [[], [-2, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, -2, -2, -2, -2, 3] Regarding the meta-word , [-2, [1], 3, 3, -2, -2, -2, -2, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, [1], -2, 3], between , [[], [-2, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, [1], -2, 3, -2, -2, -2, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, 3, 3], between , [[], [-2, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, -2, -2, -2, -2, 3] Regarding the meta-word , [[1], -2, 3, 3, -2, -2, -2, -2, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, [1]], between , [[], [-2, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, [1], -2, 3, [1], -2, -2, -2, -2, 3] Regarding the meta-word , [-2, [1], 3, [1], -2, 3, [1], -2, -2, -2, -2, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, [1], -2, 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, [1]], between , [[], [-2, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, -2, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[3, -2, [1], 3], [3, [1], -2, 3]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[3, -2, [1], 3], [3, [1], -2, 3]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, 3, 3], [3, -2, 3, -2, 3], [3, 3, -2, -2, 3], [3, -2, -2, [1], 3, [1], -2, 3], [3, -2, [1], 3, [1], -2, -2, 3]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, 3, 3], [3, -2, 3, -2, 3], [3, 3, -2, -2, 3]} Let's examine them one by one Let's consider , [3, -2, -2, 3, 3] . and enclose it inside , [[], [-2, -2, -2, -2, 3]], getting the word [3, -2, -2, 3, 3, -2, -2, -2, -2, 3] Regarding the meta-word , [3, -2, -2, 3, 3, -2, -2, -2, -2, 3], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, 3, -2, 3] . and enclose it inside , [[], [-2, -2, -2, -2, 3]], getting the word [3, -2, 3, -2, 3, -2, -2, -2, -2, 3] Regarding the meta-word , [3, -2, 3, -2, 3, -2, -2, -2, -2, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, 3, -2, -2, 3] . and enclose it inside , [[], [-2, -2, -2, -2, 3]], getting the word [3, 3, -2, -2, 3, -2, -2, -2, -2, 3] Regarding the meta-word , [3, 3, -2, -2, 3, -2, -2, -2, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 25 Lemma Number, 26 The vertex, [[-2, 3, -2, 3], [3, 3, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, -2, 3, w, 3, 3, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, -2, 3], [3, 3, -2, 3], equals , 9 The word w must have sum , -9 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-9], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, [1], -2], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, -2, [1], -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, -2, [1], -2, -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, [1], -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, [1], -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1], -2], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, [1], -2, -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, [1], -2, -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1], -2], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1], -2], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2], between , [[-2, 3, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 26 Lemma Number, 27 The vertex, [[], [3, 3, -2, -2, -2]], is a clone of vertex, [[], []] Proof: The potential mishaps are {[[1, 4], [[6], 3, [-9]]], [[1, 5], [[4], 3, [-7]]], [[1, 6], [[2], 3, [-5]]]} examining the , 1, -th mishap whose potential factorization is , [[6], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], []] In the first crack, we have to insert all the good words that sum up to , 6 which according to , CrucialLemma[6], are {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[4], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], []] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[2], 3, [-5]] It has three parts The List of Fixed Words is, [[], [3], []] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -5 which according to , CrucialLemma[-5], are {[-2, -2, -2, [1]], [-2, -2, [1], -2], [-2, [1], -2, -2], [[1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 27 Lemma Number, 28 The vertex, [[-2, 3, -2, -2, 3], [3, 3, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, -2, -2, 3, w, 3, 3, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, -2, -2, 3], [3, 3, -2, 3], equals , 7 The word w must have sum , -7 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-7], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, [1]], between , [[-2, 3, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2], between , [[-2, 3, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2], between , [[-2, 3, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, -2, [1], -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, -2, [1], -2, -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1]], between , [[-2, 3, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1]], between , [[-2, 3, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1]], between , [[-2, 3, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2], between , [[-2, 3, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, [1], -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, [1], -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2], between , [[-2, 3, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2], between , [[-2, 3, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, [1], -2, -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, [1], -2, -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 28 Lemma Number, 29 The vertex, [[-2, -2, -2], [3, 3]], is a clone of vertex, [[], []] Proof: The potential mishaps are {} The potential mishaps are {} This concludes the proof of Lemma Number , 29 Lemma Number, 30 The vertex, [[-2, 3, -2, -2, -2, -2], [3, 3, -2]], is a clone of vertex, [[], [3, -2, -2, -2]] Proof: The potential mishaps are {[[2, 7], [[5], -2]], [[7, 10], [[9], 3, [-9]]]} examining the , 1, -th mishap whose potential factorization is , [[5], -2] It has two parts The List of Fixed Words is, [[], [-2, 3, -2, -2, -2]] In the crack, we have to insert all the good words that sum up to , 5 which according to , CrucialLemma[5], are {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Since this is empty there are no minimal counterexamples coming from this pa\ rticular potential mishap examining the , 2, -th mishap whose potential factorization is , [[9], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 9 which according to , CrucialLemma[9], are {[3, 3, -2, [1], 3, [1]], [3, -2, 3, -2, 3, 3, [1]], [3, 3, 3], [-2, 3, -2, 3, 3, -2, 3, 3], [3, [1], -2, 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3, 3, -2, [1], 3], [-2, [1], 3, [1], -2, 3, 3, -2, 3, [1]], [-2, [1], 3, 3, -2, 3, 3], [-2, [1], 3, 3, -2, 3, -2, 3, 3, -2, [1]], [-2, [1], 3, 3, -2, 3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1], 3, [1], -2, 3], [-2, [1], 3, 3, -2, [1], 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, 3, [1]], [-2, [1], 3, 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2, 3, 3, [1], -2], [3, -2, [1], 3, [1], -2, 3, 3, -2, [1]], [3, -2, [1], 3, [1], -2, 3, 3, [1], -2], [3, 3, -2, 3, -2, 3, [1]], [3, 3, -2, 3, 3, -2, [1]], [3, 3, -2, 3, 3, [1], -2], [3, -2, 3, 3, 3, -2, [1]], [3, -2, 3, 3, 3, [1], -2], [3, -2, 3, 3, -2, [1], 3], [3, -2, 3, 3, -2, 3, [1]], [3, -2, 3, 3, [1], -2, 3], [3, 3, [1], -2, 3, [1]], [-2, 3, 3, -2, 3, -2, 3, 3], [-2, [1], 3, [1], -2, 3, 3, [1], -2, 3], [[1], -2, 3, 3, -2, 3, -2, 3, 3, [1], -2], [[1], -2, 3, 3, -2, 3, 3], [[1], -2, 3, 3, -2, [1], 3, 3, -2, [1]], [[1], -2, 3, 3, -2, [1], 3, 3, [1], -2], [[1], -2, 3, 3, -2, [1], 3, [1], -2, 3], [[1], -2, 3, 3, -2, 3, -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2, 3, 3, [1], -2], [[1], -2, 3, 3, 3, [1]], [-2, 3, 3, 3, [1], -2, 3], [-2, 3, 3, -2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, -2, 3, 3, [1]], [-2, 3, 3, 3, -2, 3, [1]], [-2, 3, 3, -2, [1], 3, 3], [-2, 3, 3, 3, -2, [1], 3], [-2, 3, 3, [1], -2, 3, 3], [-2, 3, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, 3, [1], -2], [-2, 3, -2, 3, 3, -2, [1], 3, 3, -2, [1]], [-2, 3, -2, 3, 3, -2, [1], 3, 3, [1], -2], [-2, 3, -2, 3, 3, [1], -2, 3, 3, [1], -2], [-2, 3, -2, 3, 3, [1], -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, 3, [1]], [-2, 3, -2, 3, 3, -2, 3, -2, 3, 3, [1], -2], [-2, 3, -2, 3, 3, -2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, 3, -2, 3, 3, -2, [1]], [3, -2, [1], 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[6], 3, [-6]]], [[1, 4], [[4], 3, [-4]]], [[1, 5], [[2], 3, [-2]]]} examining the , 1, -th mishap whose potential factorization is , [[6], 3, [-6]] It has three parts The List of Fixed Words is, [[-2, 3, -2, -2, -2, -2], [3], [3, 3, -2]] In the first crack, we have to insert all the good words that sum up to , 6 which according to , CrucialLemma[6], are {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[4], 3, [-4]] It has three parts The List of Fixed Words is, [[-2, 3, -2, -2, -2, -2], [3], [3, 3, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[2], 3, [-2]] It has three parts The List of Fixed Words is, [[-2, 3, -2, -2, -2, -2], [3], [3, 3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -2 which according to , CrucialLemma[-2], are {[-2], [[1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 30 Lemma Number, 31 The vertex, [[3, 3], [3, 3, -2, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, 3, w, 3, 3, -2, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, 3], [3, 3, -2, 3, -2], equals , 11 The word w must have sum , -11 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-11], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, -2, [1], -2], [-2, -2, -2, -2, [1], -2, -2], [-2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, -2, [1], -2, -2, -2, -2], [-2, [1], -2, -2, -2, -2, -2], [-2, [1], -2, -2, -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2], [[1], -2, -2, -2, -2, -2, [1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, [1], -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, -2, [1], -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, -2, 3, 3, -2, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2, [1], -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2, [1], -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2 ] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[-2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[-2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, -2, -2, -2, 3, -2, -2, -2], [-2, -2, -2, 3, -2, -2, -2, -2], [-2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[-2, -2, -2, -2, 3, -2, -2, -2], [-2, -2, -2, 3, -2, -2, -2, -2], [-2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's examine them one by one Let's consider , [-2, -2, -2, -2, 3, -2, -2, -2] . and enclose it inside , [[3, 3], [3, 3, -2, 3, -2]], getting the word [3, 3, -2, -2, -2, -2, 3, -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, -2, 3, -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, 3, -2, -2, -2, -2] . and enclose it inside , [[3, 3], [3, 3, -2, 3, -2]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1]] . and enclose it inside , [[3, 3], [3, 3, -2, 3, -2]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]] . and enclose it inside , [[3, 3], [3, 3, -2, 3, -2]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2] . and enclose it inside , [[3, 3], [3, 3, -2, 3, -2]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2, 3, -2] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, -2], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, -2], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, -2], [3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]] } Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 31 Lemma Number, 32 The vertex, [[-2], [3, 3, -2, 3, -2]], is a clone of vertex, [[], [3]] Proof: The potential mishaps are {[[2, 5], [[3], 3, [-9]]], [[2, 7], [[4], 3, [-10]]]} examining the , 1, -th mishap whose potential factorization is , [[3], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[4], 3, [-10]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -10 which according to , CrucialLemma[-10], are {[-2, -2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [-2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1], -2], [-2, -2, [1], -2, -2, [1], -2, -2], [-2, [1], -2, -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, -2, [1], -2], [-2, [1], -2, -2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2], [[1], -2, -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1], -2, -2], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2], [-2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 32 Lemma Number, 33 The vertex, [[], [3, -2, -2, 3, -2, 3, -2, -2]], is a clone of vertex, [[], [3, -2, -2]] Proof: The potential mishaps are {[[1, 3], [[4], 3, [-6]]], [[1, 4], [[2], 3, [-4]]], [[1, 6], [[3], 3, [-5]]], [[1, 8], [[4], 3, [-6]]], [[1, 9], [[2], 3, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[2], 3, [-4]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[3], 3, [-5]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -5 which according to , CrucialLemma[-5], are {[-2, -2, -2, [1]], [-2, -2, [1], -2], [-2, [1], -2, -2], [[1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 4, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 5, -th mishap whose potential factorization is , [[2], 3, [-4]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[4], 3, [-6]]], [[1, 4], [[2], 3, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2, 3, -2, 3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[2], 3, [-4]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2, 3, -2, 3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 33 Lemma Number, 34 The vertex, [[-2], [3, 3, -2, -2]], is a clone of vertex, [[], []] Proof: The potential mishaps are {[[2, 5], [[6], 3, [-9]]], [[2, 6], [[4], 3, [-7]]]} examining the , 1, -th mishap whose potential factorization is , [[6], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], []] In the first crack, we have to insert all the good words that sum up to , 6 which according to , CrucialLemma[6], are {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[4], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], []] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 34 Lemma Number, 35 The vertex, [[3, -2, -2, 3], [3, 3, -2, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, 3, w, 3, 3, -2, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, 3], [3, 3, -2, -2], equals , 4 The word w must have sum , -4 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-4], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} Let's examine them one at a time Enclosing , [-2, -2], between , [[3, -2, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, 3, 3, -2, -2] Regarding the meta-word , [3, -2, -2, 3, -2, -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, [1], 3, 3, -2, -2] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, [1], 3, 3, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, [1], 3, 3, -2, -2] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, [1], 3, 3, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, [1], -2, 3, 3, -2, -2] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, [1], -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 35 Lemma Number, 36 The vertex, [[3, 3], [3, 3, -2, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, 3, w, 3, 3, -2, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, 3], [3, 3, -2, -2], equals , 8 The word w must have sum , -8 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-8], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, 3, 3, -2, -2] Regarding the meta-word , [3, 3, -2, -2, -2, -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, -2] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, -2] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, -2] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2, -2], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2, -2] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2, -2] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[-2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[-2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1]]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]] . and enclose it inside , [[3, 3], [3, 3, -2, -2]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, -2] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [-2, -2, -2, 3, -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, 3, -2, -2]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3, -2, -2] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3, -2, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 36 Lemma Number, 37 The vertex, [[3, -2, -2, 3], [3, -2, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, 3, w, 3, -2, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, 3], [3, -2, -2, 3], equals , 4 The word w must have sum , -4 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-4], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} Let's examine them one at a time Enclosing , [-2, -2], between , [[3, -2, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [3, -2, -2, 3, -2, -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1]], between , [[3, -2, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, [1], 3, -2, -2, 3] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, [1], 3, -2, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, [1], 3, -2, -2, 3] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, [1], 3, -2, -2, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, [1], -2, 3, -2, -2, 3] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, [1], -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 37 Lemma Number, 38 The vertex, [[-2, -2, 3, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, -2, 3, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, -2, 3, 3], [3, 3], equals , 8 The word w must have sum , -8 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-8], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, -2, -2, -2, 3, 3] Regarding the meta-word , [-2, -2, 3, 3, -2, -2, -2, -2, 3, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1]], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, -2, [1]], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, -2, 3, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, -2, 3, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1]], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, -2, 3, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, -2, 3, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [-2, -2, 3, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1]], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, -2, 3, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, -2, 3, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [-2, -2, 3, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, [1], -2, -2, -2], between , [[-2, -2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3] Regarding the meta-word , [-2, -2, 3, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3], we have that the factor from position, 3, to position , 11, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[-2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[-2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1]]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]] . and enclose it inside , [[-2, -2, 3, 3], [3, 3]], getting the word [-2, -2, 3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [-2, -2, -2, 3, -2, -2, -2, [1]] . and enclose it inside , [[-2, -2, 3, 3], [3, 3]], getting the word [-2, -2, 3, 3, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, -2, 3, 3, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 38 Lemma Number, 39 The vertex, [[-2, 3, -2, -2, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, -2, -2, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, -2, -2, 3], [3, 3], equals , 6 The word w must have sum , -6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, [1], -2, -2, [1]], between , [[-2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1]], between , [[-2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2], between , [[-2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1]], between , [[-2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2], between , [[-2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2], between , [[-2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2], between , [[-2, 3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, 3, -2, -2, -2, 3, 3] Regarding the meta-word , [-2, 3, -2, -2, 3, -2, -2, -2, 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 39 Lemma Number, 40 The vertex, [[], [-2, -2, -2, 3, -2, 3, -2, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [w, -2, -2, -2, 3, -2, 3, -2, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [], [-2, -2, -2, 3, -2, 3, -2, -2], equals , -6 The word w must have sum , 6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} Let's examine them one at a time Enclosing , [3, 3], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, [1], -2, 3, [1]], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, [1], -2, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [3, [1], -2, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, 3, -2, [1]], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [-2, 3, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, 3, [1], -2], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [-2, 3, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, [1], 3], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], 3, -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [-2, 3, 3, -2, [1], 3, -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 6, to position , 11, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, 3, [1]], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [-2, 3, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, [1], -2, 3], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, 3, -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [-2, 3, 3, [1], -2, 3, -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 6, to position , 11, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, [1]], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, [1], 3, [1]], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, [1], 3, [1], -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [3, -2, [1], 3, [1], -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, 3, -2, [1]], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [3, -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, 3, [1], -2], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [3, -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, [1]], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [-2, [1], 3, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, 3, -2, [1]], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, [1], -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [-2, [1], 3, [1], -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 7, to position , 11, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, 3, [1], -2], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, [1], -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [-2, [1], 3, [1], -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 7, to position , 11, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, -2, 3, [1]], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [3, -2, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, [1]], between , [[], [-2, -2, -2, 3, -2, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2] Regarding the meta-word , [[1], -2, 3, 3, [1], -2, -2, -2, 3, -2, 3, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 40 Lemma Number, 41 The vertex, [[-2, 3, -2, -2, -2, 3], [3, 3, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, -2, -2, -2, 3, w, 3, 3, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, -2, -2, -2, 3], [3, 3, -2, 3], equals , 5 The word w must have sum , -5 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-5], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, [1]], [-2, -2, [1], -2], [-2, [1], -2, -2], [[1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2, -2, -2, [1]], between , [[-2, 3, -2, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2], between , [[-2, 3, -2, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, -2, -2, [1], -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, -2, 3, -2, -2, [1], -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2], between , [[-2, 3, -2, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, -2, [1], -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, -2, 3, -2, [1], -2, -2, 3, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2], between , [[-2, 3, -2, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, [1], -2, -2, -2, 3, 3, -2, 3] Regarding the meta-word , [-2, 3, -2, -2, -2, 3, [1], -2, -2, -2, 3, 3, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1]], between , [[-2, 3, -2, -2, -2, 3], [3, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[[1], -2, -2, [1], -2, -2, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[[1], -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1], -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1], -2, -2, [1]]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1], -2, -2, [1]] . and enclose it inside , [[-2, 3, -2, -2, -2, 3], [3, 3, -2, 3]], getting the word [-2, 3, -2, -2, -2, 3, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, -2, 3, -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1]], [3, -2, -2, -2, 3, -2, -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 41 Lemma Number, 42 The vertex, [[-2, 3, -2, -2, -2, -2], [3, 3, -2, 3]], is a clone of vertex, [[], []] Proof: The potential mishaps are {[[2, 7], [[5], -2, [-3]]], [[7, 10], [[6], 3, [-9]]]} examining the , 1, -th mishap whose potential factorization is , [[5], -2, [-3]] It has three parts The List of Fixed Words is, [[], [-2], []] In the first crack, we have to insert all the good words that sum up to , 5 which according to , CrucialLemma[5], are {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -3 which according to , CrucialLemma[-3], are {[-2, -2, [1]], [-2, [1], -2], [[1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[6], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], []] In the first crack, we have to insert all the good words that sum up to , 6 which according to , CrucialLemma[6], are {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 42 Lemma Number, 43 The vertex, [[], [-2, -2, -2, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [w, -2, -2, -2, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [], [-2, -2, -2, -2], equals , -8 The word w must have sum , 8 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[8], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, 3], [3, -2, 3, -2, 3, 3], [-2, 3, 3, -2, 3, 3], [3, [1], -2, 3, 3], [3, 3, 3, -2, [1]], [3, 3, 3, [1], -2], [3, 3, -2, [1], 3], [3, 3, -2, 3, [1]], [3, -2, [1], 3, 3], [3, -2, [1], 3, [1], -2, 3, [1]], [3, -2, 3, 3, [1]], [-2, [1], 3, 3, 3], [-2, [1], 3, 3, [1], -2, 3, [1]], [-2, 3, -2, 3, 3, [1], -2, 3, [1]], [-2, 3, -2, 3, 3, -2, [1], 3, [1]], [-2, 3, -2, 3, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1], 3, [1]], [-2, [1], 3, 3, -2, 3, 3, -2, [1]], [-2, [1], 3, 3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, 3, -2, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, [1]], [[1], -2, 3, 3, 3], [[1], -2, 3, 3, [1], -2, 3, [1]], [[1], -2, 3, 3, -2, [1], 3, [1]], [[1], -2, 3, 3, -2, 3, 3, -2, [1]], [[1], -2, 3, 3, -2, 3, 3, [1], -2], [[1], -2, 3, 3, -2, 3, -2, 3, [1]], [-2, 3, 3, 3, [1]], [-2, 3, 3, [1], -2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3, 3, -2, [1]], [-2, 3, 3, -2, [1], 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3, [1], -2, 3], [-2, 3, 3, -2, 3, -2, 3, 3, -2, [1]], [-2, 3, 3, -2, 3, -2, 3, 3, [1], -2], [-2, 3, -2, 3, 3, -2, 3, -2, 3, [1]]} Let's examine them one at a time Enclosing , [3, 3, [1], -2, 3], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, 3, -2, -2, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, 3, -2, 3, 3, 3], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, 3, -2, -2, -2, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, 3, -2, -2, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, -2, 3, 3], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, 3, 3, -2, -2, -2, -2] Regarding the meta-word , [3, -2, 3, -2, 3, 3, -2, -2, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, 3, 3], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, 3, 3, -2, -2, -2, -2] Regarding the meta-word , [-2, 3, 3, -2, 3, 3, -2, -2, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, [1], -2, 3, 3], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, [1], -2, 3, 3, -2, -2, -2, -2] Regarding the meta-word , [3, [1], -2, 3, 3, -2, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, 3, 3, -2, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, 3, -2, [1], -2, -2, -2, -2] Regarding the meta-word , [3, 3, 3, -2, [1], -2, -2, -2, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, 3, 3, [1], -2], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, 3, [1], -2, -2, -2, -2, -2] Regarding the meta-word , [3, 3, 3, [1], -2, -2, -2, -2, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, 3, -2, [1], 3], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], 3, -2, -2, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [3, 3, -2, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [3, 3, -2, 3, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, [1], 3, 3], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, [1], 3, 3, -2, -2, -2, -2] Regarding the meta-word , [3, -2, [1], 3, 3, -2, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, [1], 3, [1], -2, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, [1], 3, [1], -2, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [3, -2, [1], 3, [1], -2, 3, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 12, is a mishap, being , [3, [1], -2, 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [3, -2, 3, 3, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, 3], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, 3, -2, -2, -2, -2] Regarding the meta-word , [-2, [1], 3, 3, 3, -2, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, [1], -2, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, [1], -2, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, [1], 3, 3, [1], -2, 3, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 12, is a mishap, being , [3, [1], -2, 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, [1], -2, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, [1], -2, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, [1], -2, 3, [1], -2, -2, -2, -2], we have that the factor from position, 5, to position , 13, is a mishap, being , [3, [1], -2, 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, -2, [1], 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, -2, [1], 3, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, -2, [1], 3, [1], -2, -2, -2, -2], we have that the factor from position, 5, to position , 13, is a mishap, being , [3, -2, [1], 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, -2, 3, 3, -2, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, -2, 3, 3, -2, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, -2, 3, 3, -2, [1], -2, -2, -2, -2], we have that the factor from position, 5, to position , 14, is a mishap, being , [3, -2, 3, 3, -2, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, -2, 3, 3, [1], -2], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, -2, 3, 3, [1], -2, -2, -2, -2, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, -2, 3, 3, [1], -2, -2, -2, -2, -2], we have that the factor from position, 5, to position , 14, is a mishap, being , [3, -2, 3, 3, [1], -2, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, -2, [1], 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, -2, [1], 3, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, [1], 3, 3, -2, [1], 3, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 12, is a mishap, being , [3, -2, [1], 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, -2, 3, 3, -2, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, -2, 3, 3, -2, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, [1], 3, 3, -2, 3, 3, -2, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 13, is a mishap, being , [3, -2, 3, 3, -2, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, -2, 3, 3, [1], -2], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, -2, 3, 3, [1], -2, -2, -2, -2, -2] Regarding the meta-word , [-2, [1], 3, 3, -2, 3, 3, [1], -2, -2, -2, -2, -2], we have that the factor from position, 4, to position , 13, is a mishap, being , [3, -2, 3, 3, [1], -2, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, -2, 3, -2, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, -2, 3, -2, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, [1], 3, 3, -2, 3, -2, 3, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 13, is a mishap, being , [3, -2, 3, -2, 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, [1], -2, 3, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, [1], 3, [1], -2, 3, 3, [1], -2, -2, -2, -2], we have that the factor from position, 7, to position , 11, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, 3], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, 3, -2, -2, -2, -2] Regarding the meta-word , [[1], -2, 3, 3, 3, -2, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, [1], -2, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, [1], -2, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [[1], -2, 3, 3, [1], -2, 3, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 12, is a mishap, being , [3, [1], -2, 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, -2, [1], 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, -2, [1], 3, [1], -2, -2, -2, -2] Regarding the meta-word , [[1], -2, 3, 3, -2, [1], 3, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 12, is a mishap, being , [3, -2, [1], 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, -2, 3, 3, -2, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, -2, 3, 3, -2, [1], -2, -2, -2, -2] Regarding the meta-word , [[1], -2, 3, 3, -2, 3, 3, -2, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 13, is a mishap, being , [3, -2, 3, 3, -2, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, -2, 3, 3, [1], -2], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, -2, 3, 3, [1], -2, -2, -2, -2, -2] Regarding the meta-word , [[1], -2, 3, 3, -2, 3, 3, [1], -2, -2, -2, -2, -2], we have that the factor from position, 4, to position , 13, is a mishap, being , [3, -2, 3, 3, [1], -2, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, -2, 3, -2, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, -2, 3, -2, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [[1], -2, 3, 3, -2, 3, -2, 3, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 13, is a mishap, being , [3, -2, 3, -2, 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, 3, 3, 3, [1], -2, -2, -2, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, [1], -2, 3, 3, -2, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, 3, 3, -2, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, 3, 3, [1], -2, 3, 3, -2, [1], -2, -2, -2, -2], we have that the factor from position, 7, to position , 11, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, [1], -2, 3, 3, [1], -2], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, 3, 3, [1], -2, -2, -2, -2, -2] Regarding the meta-word , [-2, 3, 3, [1], -2, 3, 3, [1], -2, -2, -2, -2, -2], we have that the factor from position, 7, to position , 11, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, [1], 3, 3, -2, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], 3, 3, -2, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, 3, 3, -2, [1], 3, 3, -2, [1], -2, -2, -2, -2], we have that the factor from position, 7, to position , 11, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, [1], 3, 3, [1], -2], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], 3, 3, [1], -2, -2, -2, -2, -2] Regarding the meta-word , [-2, 3, 3, -2, [1], 3, 3, [1], -2, -2, -2, -2, -2], we have that the factor from position, 7, to position , 11, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, [1], 3, [1], -2, 3], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], 3, [1], -2, 3, -2, -2, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, 3, 3, -2, 3, -2, 3, 3, -2, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, 3, -2, 3, 3, -2, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, 3, 3, -2, 3, -2, 3, 3, -2, [1], -2, -2, -2, -2], we have that the factor from position, 5, to position , 14, is a mishap, being , [3, -2, 3, 3, -2, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, 3, -2, 3, 3, [1], -2], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, 3, -2, 3, 3, [1], -2, -2, -2, -2, -2] Regarding the meta-word , [-2, 3, 3, -2, 3, -2, 3, 3, [1], -2, -2, -2, -2, -2], we have that the factor from position, 5, to position , 14, is a mishap, being , [3, -2, 3, 3, [1], -2, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, -2, 3, -2, 3, [1]], between , [[], [-2, -2, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, -2, 3, -2, 3, [1], -2, -2, -2, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, -2, 3, -2, 3, [1], -2, -2, -2, -2], we have that the factor from position, 5, to position , 14, is a mishap, being , [3, -2, 3, -2, 3, [1], -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[3, 3, [1], -2, 3], [3, 3, -2, [1], 3], [-2, 3, 3, -2, [1], 3, [1], -2, 3]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[3, 3, [1], -2, 3], [3, 3, -2, [1], 3], [-2, 3, 3, -2, [1], 3, [1], -2, 3]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, 3, -2, -2, 3, 3], [3, 3, -2, 3, -2, 3], [3, 3, 3, -2, -2, 3], [3, 3, -2, [1], 3, [1], -2, -2, 3], [3, 3, -2, -2, [1], 3, [1], -2, 3], [-2, 3, 3, -2, 3, -2, 3, [1], -2, 3], [-2, 3, 3, -2, -2, 3, 3, [1], -2, 3], [-2, 3, 3, -2, -2, [1], 3, [1], -2, 3, [1], -2, 3]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, 3, -2, -2, 3, 3], [3, 3, -2, 3, -2, 3], [3, 3, 3, -2, -2, 3]} Let's examine them one by one Let's consider , [3, 3, -2, -2, 3, 3] . and enclose it inside , [[], [-2, -2, -2, -2]], getting the word [3, 3, -2, -2, 3, 3, -2, -2, -2, -2] Regarding the meta-word , [3, 3, -2, -2, 3, 3, -2, -2, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, 3, -2, 3, -2, 3] . and enclose it inside , [[], [-2, -2, -2, -2]], getting the word [3, 3, -2, 3, -2, 3, -2, -2, -2, -2] Regarding the meta-word , [3, 3, -2, 3, -2, 3, -2, -2, -2, -2], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, 3, 3, -2, -2, 3] . and enclose it inside , [[], [-2, -2, -2, -2]], getting the word [3, 3, 3, -2, -2, 3, -2, -2, -2, -2] Regarding the meta-word , [3, 3, 3, -2, -2, 3, -2, -2, -2, -2], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 43 Lemma Number, 44 The vertex, [[3, -2, 3], [3, -2, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, 3, w, 3, -2, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, 3], [3, -2, -2, 3], equals , 6 The word w must have sum , -6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, [1], 3, -2, -2, 3] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, [1], 3, -2, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1]], between , [[3, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, [1], 3, -2, -2, 3] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, [1], 3, -2, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2], between , [[3, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, [1], -2, 3, -2, -2, 3] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, [1], -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, [1], 3, -2, -2, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, [1], 3, -2, -2, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, [1], -2, 3, -2, -2, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, [1], -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, 3, -2, -2, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2], between , [[3, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [3, -2, 3, -2, -2, -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 44 Lemma Number, 45 The vertex, [[], [3, -2, 3, -2, 3, -2, -2]], is a clone of vertex, [[], [3, -2]] Proof: The potential mishaps are {[[1, 3], [[2], 3, [-6]]], [[1, 5], [[3], 3, [-7]]], [[1, 7], [[4], 3, [-8]]], [[1, 8], [[2], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[3], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[4], 3, [-8]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -8 which according to , CrucialLemma[-8], are {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 4, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[2], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, 3, -2, 3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 45 Lemma Number, 46 The vertex, [[3, -2, 3], [3, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, 3, w, 3, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, 3], [3, 3, -2], equals , 8 The word w must have sum , -8 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-8], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, -2, 3, 3, -2] Regarding the meta-word , [3, -2, 3, -2, -2, -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2], between , [[3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 46 Lemma Number, 47 The vertex, [[-2], [3, 3, -2, -2, 3]], is a clone of vertex, [[], [3]] Proof: The potential mishaps are {[[2, 5], [[3], 3, [-9]]], [[2, 6], [[1], 3, [-7]]]} examining the , 1, -th mishap whose potential factorization is , [[3], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[1], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 1 which according to , CrucialLemma[1], are {[[1]]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 47 Lemma Number, 48 The vertex, [[-2, -2, 3, -2, -2, -2, -2], [3, 3]], is a clone of vertex, [[], [3, -2, -2, -2]] Proof: The potential mishaps are {[[3, 8], [[5], -2]]} examining the , 1, -th mishap whose potential factorization is , [[5], -2] It has two parts The List of Fixed Words is, [[], [-2, 3, -2, -2, -2]] In the crack, we have to insert all the good words that sum up to , 5 which according to , CrucialLemma[5], are {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Since this is empty there are no minimal counterexamples coming from this pa\ rticular potential mishap The potential mishaps are {[[1, 5], [[2], 3, [-2]]], [[1, 3], [[6], 3, [-6]]], [[1, 4], [[4], 3, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-2]] It has three parts The List of Fixed Words is, [[-2, -2, 3, -2, -2, -2, -2], [3], [3, 3]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -2 which according to , CrucialLemma[-2], are {[-2], [[1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[6], 3, [-6]] It has three parts The List of Fixed Words is, [[-2, -2, 3, -2, -2, -2, -2], [3], [3, 3]] In the first crack, we have to insert all the good words that sum up to , 6 which according to , CrucialLemma[6], are {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[4], 3, [-4]] It has three parts The List of Fixed Words is, [[-2, -2, 3, -2, -2, -2, -2], [3], [3, 3]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 48 Lemma Number, 49 The vertex, [[3, -2, 3], [3, 3, -2, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, 3, w, 3, 3, -2, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, 3], [3, 3, -2, 3, -2], equals , 9 The word w must have sum , -9 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-9], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, -2, -2, -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, -2, -2, -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, [1], -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, -2, -2, -2, [1], -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1], -2], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1], -2], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 49 Lemma Number, 50 The vertex, [[3, -2, 3], [3, 3, -2, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, 3, w, 3, 3, -2, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, 3], [3, 3, -2, -2], equals , 6 The word w must have sum , -6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3, -2, -2] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3, -2, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3, -2, -2] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3, -2, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2], between , [[3, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3, -2, -2] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3, -2, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3, -2, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3, -2, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3, -2, -2] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2], between , [[3, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, 3, 3, -2, -2] Regarding the meta-word , [3, -2, 3, -2, -2, -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 50 Lemma Number, 51 The vertex, [[3, -2, -2, -2, 3], [3, -2, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, -2, 3, w, 3, -2, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, -2, 3], [3, -2, -2, 3], equals , 2 The word w must have sum , -2 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-2], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2], [[1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2], between , [[3, -2, -2, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, 3, -2, -2, 3] Regarding the meta-word , [3, -2, -2, -2, 3, -2, 3, -2, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, [1], 3, -2, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[[1], -2, -2, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[[1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, 3, -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1]] } Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1]]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1]] . and enclose it inside , [[3, -2, -2, -2, 3], [3, -2, -2, 3]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, [1], 3, -2, -2, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, -2, 3], [3, -2, -2, -2, 3, -2], [3, -2, -2, -2, -2, [1], 3, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, -2, 3]} Let's examine them one by one Let's consider , [3, -2, -2, -2, -2, 3] . and enclose it inside , [[3, -2, -2, -2, 3], [3, -2, -2, 3]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, 3, -2, -2, 3] Regarding the meta-word , [3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, 3, -2, -2, 3], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 51 Lemma Number, 52 The vertex, [[3, 3], [3, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, 3, w, 3, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, 3], [3, -2, 3], equals , 10 The word w must have sum , -10 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-10], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [-2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1], -2], [-2, -2, [1], -2, -2, [1], -2, -2], [-2, [1], -2, -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, -2, [1], -2], [-2, [1], -2, -2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2], [[1], -2, -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1], -2, -2], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2], [-2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, -2, 3, -2, 3] Regarding the meta-word , [3, 3, -2, -2, -2, -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, -2, [1], 3, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, [1], -2, 3, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, -2, -2, [1]], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1], -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, [1], -2, 3, -2, 3] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, [1], -2, 3, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, 3, -2, 3] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, [1], -2, 3, -2, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, [1], -2, 3, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, 3, -2, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, [1], -2, 3, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, [1], -2, 3, -2, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, 3, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, 3, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, [1], -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, 3, -2, -2, -2, -2, [1], -2, -2, [1], 3, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, -2, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, -2, -2, 3, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1], -2], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2], [-2, -2, -2, -2, 3, -2, -2, [1], -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[-2, -2, -2, 3, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1], -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's examine them one by one Let's consider , [-2, -2, -2, 3, -2, -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, -2, 3]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, -2, 3, -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, -2, 3]], getting the word [3, 3, -2, -2, -2, -2, 3, -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, 3, -2, -2, -2, -2, 3, -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, 3, -2, -2, -2, [1], -2] . and enclose it inside , [[3, 3], [3, -2, 3]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, 3, -2, 3] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, 3, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, -2, 3]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, -2, 3] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2] . and enclose it inside , [[3, 3], [3, -2, 3]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, -2, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], -2], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1], -2], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2], [3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 52 Lemma Number, 53 The vertex, [[3, -2, -2, -2, -2], [3, 3, -2, -2]], is a clone of vertex, [[], [3, -2, -2, -2]] Proof: The potential mishaps are {[[1, 6], [[5], -2]], [[6, 9], [[9], 3, [-9]]], [[6, 10], [[7], 3, [-7]]]} examining the , 1, -th mishap whose potential factorization is , [[5], -2] It has two parts The List of Fixed Words is, [[], [-2, 3, -2, -2, -2]] In the crack, we have to insert all the good words that sum up to , 5 which according to , CrucialLemma[5], are {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} Since this is empty there are no minimal counterexamples coming from this pa\ rticular potential mishap examining the , 2, -th mishap whose potential factorization is , [[9], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 9 which according to , CrucialLemma[9], are {[3, 3, -2, [1], 3, [1]], [3, -2, 3, -2, 3, 3, [1]], [3, 3, 3], [-2, 3, -2, 3, 3, -2, 3, 3], [3, [1], -2, 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3, 3, -2, [1], 3], [-2, [1], 3, [1], -2, 3, 3, -2, 3, [1]], [-2, [1], 3, 3, -2, 3, 3], [-2, [1], 3, 3, -2, 3, -2, 3, 3, -2, [1]], [-2, [1], 3, 3, -2, 3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1], 3, [1], -2, 3], [-2, [1], 3, 3, -2, [1], 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, 3, [1]], [-2, [1], 3, 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2, 3, 3, [1], -2], [3, -2, [1], 3, [1], -2, 3, 3, -2, [1]], [3, -2, [1], 3, [1], -2, 3, 3, [1], -2], [3, 3, -2, 3, -2, 3, [1]], [3, 3, -2, 3, 3, -2, [1]], [3, 3, -2, 3, 3, [1], -2], [3, -2, 3, 3, 3, -2, [1]], [3, -2, 3, 3, 3, [1], -2], [3, -2, 3, 3, -2, [1], 3], [3, -2, 3, 3, -2, 3, [1]], [3, -2, 3, 3, [1], -2, 3], [3, 3, [1], -2, 3, [1]], [-2, 3, 3, -2, 3, -2, 3, 3], [-2, [1], 3, [1], -2, 3, 3, [1], -2, 3], [[1], -2, 3, 3, -2, 3, -2, 3, 3, [1], -2], [[1], -2, 3, 3, -2, 3, 3], [[1], -2, 3, 3, -2, [1], 3, 3, -2, [1]], [[1], -2, 3, 3, -2, [1], 3, 3, [1], -2], [[1], -2, 3, 3, -2, [1], 3, [1], -2, 3], [[1], -2, 3, 3, -2, 3, -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2, 3, 3, [1], -2], [[1], -2, 3, 3, 3, [1]], [-2, 3, 3, 3, [1], -2, 3], [-2, 3, 3, -2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, -2, 3, 3, [1]], [-2, 3, 3, 3, -2, 3, [1]], [-2, 3, 3, -2, [1], 3, 3], [-2, 3, 3, 3, -2, [1], 3], [-2, 3, 3, [1], -2, 3, 3], [-2, 3, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, 3, [1], -2], [-2, 3, -2, 3, 3, -2, [1], 3, 3, -2, [1]], [-2, 3, -2, 3, 3, -2, [1], 3, 3, [1], -2], [-2, 3, -2, 3, 3, [1], -2, 3, 3, [1], -2], [-2, 3, -2, 3, 3, [1], -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, 3, [1]], [-2, 3, -2, 3, 3, -2, 3, -2, 3, 3, [1], -2], [-2, 3, -2, 3, 3, -2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, 3, -2, 3, 3, -2, [1]], [3, -2, [1], 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[7], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 7 which according to , CrucialLemma[7], are {[3, 3, [1]], [3, [1], -2, 3, 3, -2, [1]], [3, [1], -2, 3, 3, [1], -2], [3, -2, [1], 3, 3, -2, [1]], [3, -2, [1], 3, 3, [1], -2], [3, -2, [1], 3, [1], -2, 3], [3, -2, 3, -2, 3, 3, -2, [1]], [3, -2, 3, -2, 3, 3, [1], -2], [3, -2, 3, 3], [-2, 3, 3, 3], [-2, 3, 3, -2, 3, -2, 3, [1]], [-2, [1], 3, 3, 3, -2, [1]], [-2, [1], 3, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1], 3], [-2, [1], 3, 3, -2, 3, [1]], [-2, [1], 3, 3, [1], -2, 3], [-2, [1], 3, [1], -2, 3, 3], [[1], -2, 3, 3, 3, -2, [1]], [[1], -2, 3, 3, 3, [1], -2], [[1], -2, 3, 3, -2, [1], 3], [[1], -2, 3, 3, -2, 3, [1]], [[1], -2, 3, 3, [1], -2, 3], [-2, 3, 3, [1], -2, 3, [1]], [-2, 3, -2, 3, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, 3, [1], -2], [-2, 3, -2, 3, 3, -2, [1], 3], [-2, 3, -2, 3, 3, -2, 3, [1]], [-2, 3, -2, 3, 3, [1], -2, 3], [-2, 3, 3, -2, [1], 3, [1]], [-2, 3, 3, -2, 3, 3, -2, [1]], [-2, 3, 3, -2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[6], 3, [-6]]], [[1, 4], [[4], 3, [-4]]], [[1, 5], [[2], 3, [-2]]]} examining the , 1, -th mishap whose potential factorization is , [[6], 3, [-6]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, 3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 6 which according to , CrucialLemma[6], are {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[4], 3, [-4]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, 3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[2], 3, [-2]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, 3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -2 which according to , CrucialLemma[-2], are {[-2], [[1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 53 Lemma Number, 54 The vertex, [[3, -2, -2, -2, 3], [3, 3, -2, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, -2, 3, w, 3, 3, -2, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, -2, 3], [3, 3, -2, -2], equals , 2 The word w must have sum , -2 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-2], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2], [[1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2], between , [[3, -2, -2, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, 3, 3, -2, -2] Regarding the meta-word , [3, -2, -2, -2, 3, -2, 3, 3, -2, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, 3, -2, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, [1], 3, 3, -2, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[[1], -2, -2, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[[1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1]], [-2, 3, -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1]] } Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1]]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1]] . and enclose it inside , [[3, -2, -2, -2, 3], [3, 3, -2, -2]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, [1], 3, 3, -2, -2] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, -2, 3], [3, -2, -2, -2, -2, [1], 3, [1], -2], [3, -2, -2, -2, 3, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, -2, 3]} Let's examine them one by one Let's consider , [3, -2, -2, -2, -2, 3] . and enclose it inside , [[3, -2, -2, -2, 3], [3, 3, -2, -2]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, 3, 3, -2, -2] Regarding the meta-word , [3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, 3, 3, -2, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 54 Lemma Number, 55 The vertex, [[], [-2, 3, -2, -2, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [w, -2, 3, -2, -2, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [], [-2, 3, -2, -2, -2, 3], equals , -2 The word w must have sum , 2 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[2], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} Let's examine them one at a time Enclosing , [3, -2, [1]], between , [[], [-2, 3, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, [1], -2, 3, -2, -2, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [3, [1], -2], between , [[], [-2, 3, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, [1], -2, -2, 3, -2, -2, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, [1], 3], between , [[], [-2, 3, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, -2, 3, -2, -2, -2, 3] Regarding the meta-word , [-2, [1], 3, -2, 3, -2, -2, -2, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3], between , [[], [-2, 3, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, -2, 3, -2, -2, -2, 3] Regarding the meta-word , [[1], -2, 3, -2, 3, -2, -2, -2, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, [1]], between , [[], [-2, 3, -2, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, [1], -2, 3, -2, -2, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[3, -2, [1]], [3, [1], -2], [-2, 3, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[3, -2, [1]], [3, [1], -2], [-2, 3, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, 3], [-2, 3, -2, 3], [3, -2, 3, -2], [-2, 3, 3, -2], [3, 3, -2, -2], [3, -2, -2, [1], 3, [1], -2], [-2, 3, -2, [1], 3, [1], -2], [3, -2, [1], 3, [1], -2, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, 3], [-2, 3, -2, 3], [3, -2, 3, -2], [-2, 3, 3, -2], [3, 3, -2, -2] } Let's examine them one by one Let's consider , [3, -2, -2, 3] . and enclose it inside , [[], [-2, 3, -2, -2, -2, 3]], getting the word [3, -2, -2, 3, -2, 3, -2, -2, -2, 3] Regarding the meta-word , [3, -2, -2, 3, -2, 3, -2, -2, -2, 3], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, 3, -2, 3] . and enclose it inside , [[], [-2, 3, -2, -2, -2, 3]], getting the word [-2, 3, -2, 3, -2, 3, -2, -2, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, 3, -2, -2, -2, 3], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, 3, -2] . and enclose it inside , [[], [-2, 3, -2, -2, -2, 3]], getting the word [3, -2, 3, -2, -2, 3, -2, -2, -2, 3] Regarding the meta-word , [3, -2, 3, -2, -2, 3, -2, -2, -2, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, 3, 3, -2] . and enclose it inside , [[], [-2, 3, -2, -2, -2, 3]], getting the word [-2, 3, 3, -2, -2, 3, -2, -2, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, 3, -2, -2, -2, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, 3, -2, -2] . and enclose it inside , [[], [-2, 3, -2, -2, -2, 3]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, 3] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 55 Lemma Number, 56 The vertex, [[], [3, 3, -2, 3, -2, -2]], is a clone of vertex, [[], [3]] Proof: The potential mishaps are {[[1, 4], [[3], 3, [-9]]], [[1, 6], [[4], 3, [-10]]], [[1, 7], [[2], 3, [-8]]]} examining the , 1, -th mishap whose potential factorization is , [[3], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[4], 3, [-10]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -10 which according to , CrucialLemma[-10], are {[-2, -2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [-2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1], -2], [-2, -2, [1], -2, -2, [1], -2, -2], [-2, [1], -2, -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, -2, [1], -2], [-2, [1], -2, -2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2], [[1], -2, -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1], -2, -2], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2], [-2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[2], 3, [-8]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -8 which according to , CrucialLemma[-8], are {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 56 Lemma Number, 57 The vertex, [[-2, 3, -2, -2, -2, -2], [3, 3]], is a clone of vertex, [[], [3, -2, -2]] Proof: The potential mishaps are {[[2, 7], [[5], -2, [-2]]]} examining the , 1, -th mishap whose potential factorization is , [[5], -2, [-2]] It has three parts The List of Fixed Words is, [[], [-2], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 5 which according to , CrucialLemma[5], are {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -2 which according to , CrucialLemma[-2], are {[-2], [[1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[4], 3, [-6]]], [[1, 4], [[2], 3, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[-2, 3, -2, -2, -2, -2], [3], [3, 3]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[2], 3, [-4]] It has three parts The List of Fixed Words is, [[-2, 3, -2, -2, -2, -2], [3], [3, 3]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 57 Lemma Number, 58 The vertex, [[-2], [-2, -2, 3, -2, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, w, -2, -2, 3, -2, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2], [-2, -2, 3, -2, -2, 3], equals , -4 The word w must have sum , 4 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[4], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} Let's examine them one at a time Enclosing , [3, [1]], between , [[-2], [-2, -2, 3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, [1], -2, -2, 3, -2, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, 3, 3, -2, [1]], between , [[-2], [-2, -2, 3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], -2, 3, 3, -2, [1], -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, [1], -2, 3, 3, -2, [1], -2, -2, 3, -2, -2, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, [1], -2], between , [[-2], [-2, -2, 3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], -2, 3, 3, [1], -2, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, [1], -2, 3, 3, [1], -2, -2, -2, 3, -2, -2, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, -2, [1]], between , [[-2], [-2, -2, 3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, [1], 3, 3, -2, [1], -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, -2, [1], 3, 3, -2, [1], -2, -2, 3, -2, -2, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, [1], -2], between , [[-2], [-2, -2, 3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, [1], 3, 3, [1], -2, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, -2, [1], 3, 3, [1], -2, -2, -2, 3, -2, -2, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3], between , [[-2], [-2, -2, 3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, [1], 3, [1], -2, 3, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, -2, [1], 3, [1], -2, 3, -2, -2, 3, -2, -2, 3], we have that the factor from position, 7, to position , 12, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, -2, [1]], between , [[-2], [-2, -2, 3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, 3, -2, [1], -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, 3, -2, [1], -2, -2, 3, -2, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, [1], -2], between , [[-2], [-2, -2, 3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, 3, [1], -2, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, 3, [1], -2, -2, -2, 3, -2, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3], between , [[-2], [-2, -2, 3, -2, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, -2, 3, 3, -2, -2, 3, -2, -2, 3], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[3, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[3, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, 3], [3, 3, -2], [3, -2, [1], 3, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, 3], [3, 3, -2], [3, -2, [1], 3, [1], -2]} Let's examine them one by one Let's consider , [3, -2, 3] . and enclose it inside , [[-2], [-2, -2, 3, -2, -2, 3]], getting the word [-2, 3, -2, 3, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, 3, -2, -2, 3], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, 3, -2] . and enclose it inside , [[-2], [-2, -2, 3, -2, -2, 3]], getting the word [-2, 3, 3, -2, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, 3, -2, -2, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, [1], 3, [1], -2] . and enclose it inside , [[-2], [-2, -2, 3, -2, -2, 3]], getting the word [-2, 3, -2, [1], 3, [1], -2, -2, -2, 3, -2, -2, 3] Regarding the meta-word , [-2, 3, -2, [1], 3, [1], -2, -2, -2, 3, -2, -2, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 58 Lemma Number, 59 The vertex, [[-2, -2], [3, 3, -2, 3]], is a clone of vertex, [[], [3]] Proof: The potential mishaps are {[[3, 6], [[3], 3, [-9]]]} examining the , 1, -th mishap whose potential factorization is , [[3], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 59 Lemma Number, 60 The vertex, [[3, -2, -2, -2, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, -2, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, -2, 3], [3, 3], equals , 6 The word w must have sum , -6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, [1], -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, -2, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2], between , [[3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, -2, -2, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, -2, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2], between , [[3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, -2, -2, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2], between , [[3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, -2], between , [[3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, -2, -2, 3, 3] Regarding the meta-word , [3, -2, -2, -2, 3, -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[[1], -2, -2, [1], -2, -2]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[[1], -2, -2, [1], -2, -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, 3, -2, -2, [1], -2, -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2], [3, -2, -2, -2, [1], -2, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1], -2, -2]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1], -2, -2] . and enclose it inside , [[3, -2, -2, -2, 3], [3, 3]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, [1], -2, -2, 3, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, -2, 3, -2, -2], [3, -2, -2, -2, 3, -2, -2, -2], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, -2, 3, -2, -2]} Let's examine them one by one Let's consider , [3, -2, -2, -2, -2, 3, -2, -2] . and enclose it inside , [[3, -2, -2, -2, 3], [3, 3]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, -2, -2, 3, 3] Regarding the meta-word , [3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, -2, -2, 3, 3], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 60 Lemma Number, 61 The vertex, [[3, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, 3], [3, 3], equals , 12 The word w must have sum , -12 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-12], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, -2, -2], [-2, -2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2, -2, [1], -2], [-2, -2, [1], -2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, -2, -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, -2, [1], -2, -2], [-2, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2, -2, -2], [-2, -2, -2, -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2, -2, -2], [[1], -2, -2, [1], -2, -2, -2, -2, -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, -2, -2, 3, 3] Regarding the meta-word , [3, 3, -2, -2, -2, -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, -2, -2, -2, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, 3, -2, -2, -2, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, -2, [1], -2, -2, -2, [1], -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, -2, [1], -2, -2, [1], -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3] Regarding the meta-word , [3, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, -2, 3, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, -2, [1], -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, -2, 3, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, -2, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, -2, [1], -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1], -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 2, to position , 6, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, -2, 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, -2, [1], -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, -2, 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, -2, 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1], 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2, 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2], between , [[3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[-2, -2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[-2, -2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, -2, -2, 3, -2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, -2, [1], -2], [-2, -2, -2, -2, 3, -2, -2, -2, [1], -2], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, -2, [1], -2], [-2, -2, -2, 3, -2, -2, -2, [1], -2, -2], [-2, -2, -2, -2, 3, -2, -2, [1], -2, -2], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[-2, -2, -2, 3, -2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, -2, [1], -2], [-2, -2, -2, -2, 3, -2, -2, -2, [1], -2], [-2, -2, -2, 3, -2, -2, -2, [1], -2, -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2]} Let's examine them one by one Let's consider , [-2, -2, -2, 3, -2, -2, -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, 3]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, -2, 3, -2, -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, 3]], getting the word [3, 3, -2, -2, -2, -2, 3, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, 3, -2, -2, -2, -2, 3, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, 3, -2, -2, -2, -2, [1], -2] . and enclose it inside , [[3, 3], [3, 3]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, -2, 3, -2, -2, -2, [1], -2] . and enclose it inside , [[3, 3], [3, 3]], getting the word [3, 3, -2, -2, -2, -2, 3, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, 3, -2, -2, -2, -2, 3, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, 3, -2, -2, -2, [1], -2, -2] . and enclose it inside , [[3, 3], [3, 3]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]] . and enclose it inside , [[3, 3], [3, 3]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1], 3, 3] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2] . and enclose it inside , [[3, 3], [3, 3]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2, 3, 3] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2] . and enclose it inside , [[3, 3], [3, 3]], getting the word [3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, 3, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, -2, -2, [1]], [3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, -2, [1], -2], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, -2, [1], -2], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, -2, [1], -2], [3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1], -2, -2], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 61 Lemma Number, 62 The vertex, [[-2, -2], [3, -2, -2, 3, -2, 3]], is a clone of vertex, [[], [3, -2, -2]] Proof: The potential mishaps are {[[3, 5], [[4], 3, [-6]]], [[3, 6], [[2], 3, [-4]]], [[3, 8], [[3], 3, [-5]]]} examining the , 1, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[2], 3, [-4]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[3], 3, [-5]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -5 which according to , CrucialLemma[-5], are {[-2, -2, -2, [1]], [-2, -2, [1], -2], [-2, [1], -2, -2], [[1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[4], 3, [-6]]], [[1, 4], [[2], 3, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[-2, -2], [3], [3, -2, -2, 3, -2, 3]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[2], 3, [-4]] It has three parts The List of Fixed Words is, [[-2, -2], [3], [3, -2, -2, 3, -2, 3]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 62 Lemma Number, 63 The vertex, [[-2, -2], [-2, -2, -2, 3, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, -2, w, -2, -2, -2, 3, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, -2], [-2, -2, -2, 3, -2, 3], equals , -6 The word w must have sum , 6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[3, 3], [3, [1], -2, 3, [1]], [-2, 3, 3, 3, -2, [1]], [-2, 3, 3, 3, [1], -2], [-2, 3, 3, -2, [1], 3], [-2, 3, 3, -2, 3, [1]], [-2, 3, 3, [1], -2, 3], [-2, 3, -2, 3, 3, [1]], [3, -2, [1], 3, [1]], [3, -2, 3, 3, -2, [1]], [3, -2, 3, 3, [1], -2], [-2, [1], 3, 3, [1]], [-2, [1], 3, [1], -2, 3, 3, -2, [1]], [-2, [1], 3, [1], -2, 3, 3, [1], -2], [3, -2, 3, -2, 3, [1]], [[1], -2, 3, 3, [1]]} Let's examine them one at a time Enclosing , [3, 3], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, 3, -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, 3, 3, -2, -2, -2, 3, -2, 3], we have that the factor from position, 4, to position , 9, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, [1], -2, 3, [1]], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, [1], -2, 3, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, 3, [1], -2, 3, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, 3, -2, [1]], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, -2, 3, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, -2, 3, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, 3, [1], -2], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, -2, 3, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, -2, 3, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, [1], 3], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, -2, 3, 3, -2, [1], 3, -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, -2, 3, 3, -2, [1], 3, -2, -2, -2, 3, -2, 3], we have that the factor from position, 8, to position , 13, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, -2, 3, [1]], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, -2, 3, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, -2, 3, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 7, to position , 11, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3, [1], -2, 3], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, -2, 3, 3, [1], -2, 3, -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, -2, 3, 3, [1], -2, 3, -2, -2, -2, 3, -2, 3], we have that the factor from position, 8, to position , 13, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, [1]], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, -2, 3, -2, 3, 3, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, -2, 3, -2, 3, 3, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 7, to position , 11, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, [1], 3, [1]], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, [1], 3, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, 3, -2, [1], 3, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, 3, -2, [1]], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, 3, [1], -2], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, [1]], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, -2, [1], 3, 3, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, -2, [1], 3, 3, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, 3, -2, [1]], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, -2, [1], 3, [1], -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, -2, [1], 3, [1], -2, 3, 3, -2, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 9, to position , 13, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3, 3, [1], -2], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, -2, [1], 3, [1], -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, -2, [1], 3, [1], -2, 3, 3, [1], -2, -2, -2, -2, 3, -2, 3], we have that the factor from position, 9, to position , 13, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [3, -2, 3, -2, 3, [1]], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, 3, -2, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, 3, -2, 3, -2, 3, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 7, to position , 11, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, [1]], between , [[-2, -2], [-2, -2, -2, 3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, -2, [1], -2, 3, 3, [1], -2, -2, -2, 3, -2, 3] Regarding the meta-word , [-2, -2, [1], -2, 3, 3, [1], -2, -2, -2, 3, -2, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 63 Lemma Number, 64 The vertex, [[], [3, 3, -2, -2, 3, -2]], is a clone of vertex, [[], [3]] Proof: The potential mishaps are {[[1, 4], [[3], 3, [-9]]], [[1, 5], [[1], 3, [-7]]], [[1, 7], [[2], 3, [-8]]]} examining the , 1, -th mishap whose potential factorization is , [[3], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[1], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 1 which according to , CrucialLemma[1], are {[[1]]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[2], 3, [-8]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -8 which according to , CrucialLemma[-8], are {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 64 Lemma Number, 65 The vertex, [[-2, 3, -2, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, -2, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, -2, 3], [3, 3], equals , 8 The word w must have sum , -8 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-8], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, -2, -2, 3, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, -2, -2, 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [-2, 3, -2, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2], between , [[-2, 3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 65 Lemma Number, 66 The vertex, [[-2, 3, 3], [3, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, 3, w, 3, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, 3], [3, 3, -2], equals , 8 The word w must have sum , -8 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-8], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, -2, 3, 3, -2] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, -2, 3, 3, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [-2, 3, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [-2, 3, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3, -2], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3, -2], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, [1], -2, -2, -2], between , [[-2, 3, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3, -2], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[-2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[-2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1]]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]] . and enclose it inside , [[-2, 3, 3], [3, 3, -2]], getting the word [-2, 3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [-2, -2, -2, 3, -2, -2, -2, [1]] . and enclose it inside , [[-2, 3, 3], [3, 3, -2]], getting the word [-2, 3, 3, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1]]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 66 Lemma Number, 67 The vertex, [[3, -2, -2, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, 3], [3, 3], equals , 8 The word w must have sum , -8 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-8], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2], [-2, [1], -2, -2, [1], -2, -2], [[1], -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, [1], -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, -2, -2, 3, 3] Regarding the meta-word , [3, -2, -2, 3, -2, -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, -2, 3, -2, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, -2, -2, 3, -2, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2], between , [[3, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 67 Lemma Number, 68 The vertex, [[-2, 3, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, 3], [3, 3], equals , 10 The word w must have sum , -10 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-10], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [-2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1], -2], [-2, -2, [1], -2, -2, [1], -2, -2], [-2, [1], -2, -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, -2, [1], -2], [-2, [1], -2, -2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2], [[1], -2, -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1], -2, -2], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2], [-2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, -2, -2, 3, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, -2, -2, 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, [1], -2, -2, -2, [1], 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, -2, [1], -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [-2, -2, [1], -2, -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3] Regarding the meta-word , [-2, 3, 3, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, -2, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, -2, 3, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, -2, [1], -2, -2, -2, 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, -2, 3, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, -2, 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], between , [[-2, 3, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, 3, [1], -2, -2, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[-2, -2, -2, 3, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1], -2], [-2, -2, -2, -2, 3, -2, -2, [1], -2], [-2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, 3, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[-2, -2, -2, 3, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, 3, -2, -2, -2, [1]], [-2, -2, -2, 3, -2, -2, -2, [1], -2], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's examine them one by one Let's consider , [-2, -2, -2, 3, -2, -2, -2, -2, [1]] . and enclose it inside , [[-2, 3, 3], [3, 3]], getting the word [-2, 3, 3, -2, -2, -2, 3, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, 3, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, -2, 3, -2, -2, -2, [1]] . and enclose it inside , [[-2, 3, 3], [3, 3]], getting the word [-2, 3, 3, -2, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, -2, 3, -2, -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [-2, -2, -2, 3, -2, -2, -2, [1], -2] . and enclose it inside , [[-2, 3, 3], [3, 3]], getting the word [-2, 3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [-2, 3, 3, -2, -2, -2, 3, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]] . and enclose it inside , [[-2, 3, 3], [3, 3]], getting the word [-2, 3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3] Since this meta-word does NOT contain mishaps, we keep it Let's consider , [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2] . and enclose it inside , [[-2, 3, 3], [3, 3]], getting the word [-2, 3, 3, 3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, [1], -2], [3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2, -2, [1], -2, -2, -2, [1]], [3, -2, -2, -2, 3, -2, -2, -2, [1], -2, -2, [1], -2], [3, -2, -2, -2, -2, 3, -2, -2, [1], -2, -2, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps There is nothing left!, QED This concludes the proof of Lemma Number , 68 Lemma Number, 69 The vertex, [[3, -2, -2, -2, 3], [3, -2, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, -2, 3, w, 3, -2, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, -2, 3], [3, -2, 3], equals , 4 The word w must have sum , -4 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-4], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} Let's examine them one at a time Enclosing , [-2, -2], between , [[3, -2, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, -2, 3, -2, 3] Regarding the meta-word , [3, -2, -2, -2, 3, -2, -2, 3, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, [1], -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, -2, -2, -2, 3, -2, [1], -2, -2, [1], 3, -2, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, -2, [1], 3, -2, 3] Regarding the meta-word , [3, -2, -2, -2, 3, [1], -2, -2, -2, [1], 3, -2, 3], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2], between , [[3, -2, -2, -2, 3], [3, -2, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, [1], -2, 3, -2, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[[1], -2, -2, [1], -2]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[[1], -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1], -2], [-2, 3, -2, -2, [1], -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1], -2]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1], -2] . and enclose it inside , [[3, -2, -2, -2, 3], [3, -2, 3]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, [1], -2, 3, -2, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, -2, 3, -2], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2], [3, -2, -2, -2, 3, -2, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, -2, 3, -2]} Let's examine them one by one Let's consider , [3, -2, -2, -2, -2, 3, -2] . and enclose it inside , [[3, -2, -2, -2, 3], [3, -2, 3]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, -2, 3, -2, 3] Regarding the meta-word , [3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, -2, 3, -2, 3], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 69 Lemma Number, 70 The vertex, [[3, -2, -2, -2, -2], [3, -2, 3]], is a clone of vertex, [[], [3, -2, -2]] Proof: The potential mishaps are {[[6, 8], [[4], 3, [-6]]], [[1, 6], [[5], -2, [-2]]]} examining the , 1, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[5], -2, [-2]] It has three parts The List of Fixed Words is, [[], [-2], [3, -2, -2]] In the first crack, we have to insert all the good words that sum up to , 5 which according to , CrucialLemma[5], are {[-2, 3, -2, 3, 3], [3, 3, -2, [1]], [3, 3, [1], -2], [3, -2, [1], 3], [3, -2, 3, [1]], [-2, [1], 3, 3], [3, [1], -2, 3], [[1], -2, 3, 3], [-2, [1], 3, [1], -2, 3, [1]], [-2, 3, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -2 which according to , CrucialLemma[-2], are {[-2], [[1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[4], 3, [-6]]], [[1, 4], [[2], 3, [-4]]]} examining the , 1, -th mishap whose potential factorization is , [[4], 3, [-6]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, -2, 3]] In the first crack, we have to insert all the good words that sum up to , 4 which according to , CrucialLemma[4], are {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[2], 3, [-4]] It has three parts The List of Fixed Words is, [[3, -2, -2, -2, -2], [3], [3, -2, 3]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -4 which according to , CrucialLemma[-4], are {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 70 Lemma Number, 71 The vertex, [[3], [-2, 3, -2, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, w, -2, 3, -2, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3], [-2, 3, -2, 3, -2], equals , 3 The word w must have sum , -3 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-3], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, [1]], [-2, [1], -2], [[1], -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, [1]], between , [[3], [-2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, [1], -2, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, [1], -2, 3, -2, 3, -2], we have that the factor from position, 1, to position , 5, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2], between , [[3], [-2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, [1], -2, -2, 3, -2, 3, -2] Regarding the meta-word , [3, -2, [1], -2, -2, 3, -2, 3, -2], we have that the factor from position, 1, to position , 5, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2], between , [[3], [-2, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, [1], -2, -2, -2, 3, -2, 3, -2] Regarding the meta-word , [3, [1], -2, -2, -2, 3, -2, 3, -2], we have that the factor from position, 1, to position , 5, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 71 Lemma Number, 72 The vertex, [[3, -2, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, 3], [3, 3], equals , 10 The word w must have sum , -10 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-10], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, -2], [-2, -2, -2, [1], -2, -2, -2, [1]], [-2, -2, -2, [1], -2, -2, [1], -2], [-2, -2, [1], -2, -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2, [1], -2], [-2, -2, [1], -2, -2, [1], -2, -2], [-2, [1], -2, -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, -2, [1], -2], [-2, [1], -2, -2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2], [[1], -2, -2, -2, -2, -2, -2, [1]], [[1], -2, -2, -2, -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1], -2, -2], [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2], [-2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, -2, -2, 3, 3] Regarding the meta-word , [3, -2, 3, -2, -2, -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, 3, -2, -2, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2, -2, [1], -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, -2, 3, -2, -2, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [3, -2, 3, -2, -2, [1], -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3] Regarding the meta-word , [3, -2, 3, -2, [1], -2, -2, [1], -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, -2, -2, [1], 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, -2, [1], -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, -2, [1], -2, 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1], -2, -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, -2, [1], -2, -2, 3, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, -2, [1], -2, -2, 3, 3], we have that the factor from position, 3, to position , 7, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1]], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, -2, 3, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, -2, [1], -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, -2], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, -2, 3, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, -2, 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, 3, -2, -2, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1]], between , [[3, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [3, -2, 3, [1], -2, -2, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 72 Lemma Number, 73 The vertex, [[3, -2, -2, 3], [3, 3, -2, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, 3, w, 3, 3, -2, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, 3], [3, 3, -2, 3, -2], equals , 7 The word w must have sum , -7 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-7], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, 3, -2, -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, 3, -2, -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, [1], -2, -2], between , [[3, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, -2, [1], -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, 3, -2, -2, [1], -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, [1], -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, -2, [1]], between , [[3, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, -2, [1], 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2], between , [[3, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, -2, [1], -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, 3, -2, [1], -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2, [1], -2], between , [[3, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, [1], -2, -2, [1], -2, 3, 3, -2, 3, -2], we have that the factor from position, 1, to position , 9, is a mishap, being , [3, -2, -2, 3, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2], between , [[3, -2, -2, 3], [3, 3, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, 3, [1], -2, -2, -2, -2, 3, 3, -2, 3, -2] Regarding the meta-word , [3, -2, -2, 3, [1], -2, -2, -2, -2, 3, 3, -2, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 73 Lemma Number, 74 The vertex, [[-2, 3, -2, -2, -2, 3], [3, 3]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, -2, -2, -2, 3, w, 3, 3] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, -2, -2, -2, 3], [3, 3], equals , 4 The word w must have sum , -4 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-4], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} Let's examine them one at a time Enclosing , [-2, -2], between , [[-2, 3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, -2, -2, 3, 3] Regarding the meta-word , [-2, 3, -2, -2, -2, 3, -2, -2, 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1]], between , [[-2, 3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, -2, [1], -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, -2, -2, -2, 3, -2, [1], -2, -2, [1], 3, 3], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1]], between , [[-2, 3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, [1], -2, -2, -2, [1], 3, 3] Regarding the meta-word , [-2, 3, -2, -2, -2, 3, [1], -2, -2, -2, [1], 3, 3], we have that the factor from position, 6, to position , 10, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2], between , [[-2, 3, -2, -2, -2, 3], [3, 3]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, [1], -2, -2, [1], -2, 3, 3] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[[1], -2, -2, [1], -2]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[[1], -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1], -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2], [-2, 3, -2, -2, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1], -2]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1], -2] . and enclose it inside , [[-2, 3, -2, -2, -2, 3], [3, 3]], getting the word [-2, 3, -2, -2, -2, 3, 3, -2, -2, -2, [1], -2, 3, 3] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, 3, -2, -2], [3, -2, -2, -2, -2, [1], 3, [1], -2, -2], [3, -2, -2, -2, -2, 3, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, -2, 3, -2]} Let's examine them one by one Let's consider , [3, -2, -2, -2, -2, 3, -2] . and enclose it inside , [[-2, 3, -2, -2, -2, 3], [3, 3]], getting the word [-2, 3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, -2, 3, 3] Regarding the meta-word , [-2, 3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, -2, 3, 3], we have that the factor from position, 6, to position , 11, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 74 Lemma Number, 75 The vertex, [[-2, 3, -2, 3], [3, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, -2, 3, w, 3, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, -2, 3], [3, 3, -2], equals , 6 The word w must have sum , -6 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-6], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} Let's examine them one at a time Enclosing , [-2, -2, [1], -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, [1], -2, -2, [1], 3, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, -2, [1], -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1], -2], between , [[-2, 3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, -2, [1], -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, -2, [1], -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, -2, [1]], between , [[-2, 3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1], -2], between , [[-2, 3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, -2, [1], -2, 3, 3, -2], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, [1], -2, -2, -2, [1], -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2, -2], between , [[-2, 3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, [1], -2, -2, [1], -2, -2, 3, 3, -2], we have that the factor from position, 2, to position , 10, is a mishap, being , [3, -2, 3, [1], -2, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, -2, -2], between , [[-2, 3, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, -2, -2, -2, 3, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, -2, -2, -2, 3, 3, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, 3, -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning They were all kicked-out after the first prunings! Everything disapeared after the first purge! This concludes the proof of Lemma Number , 75 Lemma Number, 76 The vertex, [[-2, -2], [3, -2, 3, -2, 3]], is a clone of vertex, [[], [3, -2]] Proof: The potential mishaps are {[[3, 5], [[2], 3, [-6]]], [[3, 7], [[3], 3, [-7]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[3], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[2], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[-2, -2], [3], [3, -2, 3, -2, 3]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 76 Lemma Number, 77 The vertex, [[-2, 3, -2, -2, -2, 3], [3, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [-2, 3, -2, -2, -2, 3, w, 3, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [-2, 3, -2, -2, -2, 3], [3, 3, -2], equals , 2 The word w must have sum , -2 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-2], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2], [[1], -2, -2, [1]]} Let's examine them one at a time Enclosing , [-2], between , [[-2, 3, -2, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, -2, 3, 3, -2] Regarding the meta-word , [-2, 3, -2, -2, -2, 3, -2, 3, 3, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1]], between , [[-2, 3, -2, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, -2, -2, 3, [1], -2, -2, [1], 3, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[[1], -2, -2, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[[1], -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1]], [-2, 3, -2, -2, [1]], [-2, [1], 3, [1], -2, -2, -2, [1]] } Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1]]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1]] . and enclose it inside , [[-2, 3, -2, -2, -2, 3], [3, 3, -2]], getting the word [-2, 3, -2, -2, -2, 3, 3, -2, -2, -2, [1], 3, 3, -2] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1]]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, -2, 3], [3, -2, -2, -2, 3, -2], [3, -2, -2, -2, -2, [1], 3, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, -2, 3]} Let's examine them one by one Let's consider , [3, -2, -2, -2, -2, 3] . and enclose it inside , [[-2, 3, -2, -2, -2, 3], [3, 3, -2]], getting the word [-2, 3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, 3, 3, -2] Regarding the meta-word , [-2, 3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, 3, 3, -2], we have that the factor from position, 6, to position , 11, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 77 Lemma Number, 78 The vertex, [[], [3, 3, -2, -2, -2, 3]], is a clone of vertex, [[], [3]] Proof: The potential mishaps are {[[1, 4], [[3], 3, [-9]]], [[1, 5], [[1], 3, [-7]]], [[1, 6], [[-1], 3, [-5]]]} examining the , 1, -th mishap whose potential factorization is , [[3], 3, [-9]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -9 which according to , CrucialLemma[-9], are {[-2, -2, -2, -2, -2, [1]], [-2, -2, -2, -2, [1], -2], [-2, -2, -2, [1], -2, -2], [-2, -2, [1], -2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, [1], -2, -2, -2, [1]], [-2, -2, [1], -2, -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1], -2], [-2, [1], -2, -2, -2, -2], [[1], -2, -2, -2, -2, -2], [[1], -2, -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, -2, -2, [1]], [-2, [1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2, [1], -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[1], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , 1 which according to , CrucialLemma[1], are {[[1]]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[-1], 3, [-5]] It has three parts The List of Fixed Words is, [[], [3], [3]] In the first crack, we have to insert all the good words that sum up to , -1 which according to , CrucialLemma[-1], are {[-2, [1]], [[1], -2]} In the second crack, we have to insert all the good words that sum up to , -5 which according to , CrucialLemma[-5], are {[-2, -2, -2, [1]], [-2, -2, [1], -2], [-2, [1], -2, -2], [[1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {} This concludes the proof of Lemma Number , 78 Lemma Number, 79 The vertex, [[], [-2, -2, 3, -2, -2, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [w, -2, -2, 3, -2, -2, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [], [-2, -2, 3, -2, -2, 3, -2], equals , -4 The word w must have sum , 4 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[4], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[3, [1]], [[1], -2, 3, 3, -2, [1]], [[1], -2, 3, 3, [1], -2], [-2, [1], 3, 3, -2, [1]], [-2, [1], 3, 3, [1], -2], [-2, [1], 3, [1], -2, 3], [-2, 3, -2, 3, 3, -2, [1]], [-2, 3, -2, 3, 3, [1], -2], [-2, 3, 3]} Let's examine them one at a time Enclosing , [3, [1]], between , [[], [-2, -2, 3, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, [1], -2, -2, 3, -2, -2, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Enclosing , [[1], -2, 3, 3, -2, [1]], between , [[], [-2, -2, 3, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, -2, [1], -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [[1], -2, 3, 3, -2, [1], -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, 3, 3, [1], -2], between , [[], [-2, -2, 3, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [[1], -2, 3, 3, [1], -2, -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [[1], -2, 3, 3, [1], -2, -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, -2, [1]], between , [[], [-2, -2, 3, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, -2, [1], -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [-2, [1], 3, 3, -2, [1], -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, 3, [1], -2], between , [[], [-2, -2, 3, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, 3, [1], -2, -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [-2, [1], 3, 3, [1], -2, -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], 3, [1], -2, 3], between , [[], [-2, -2, 3, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, [1], 3, [1], -2, 3, -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [-2, [1], 3, [1], -2, 3, -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 6, to position , 11, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, -2, [1]], between , [[], [-2, -2, 3, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, -2, [1], -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, -2, [1], -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, -2, [1], -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, -2, 3, 3, [1], -2], between , [[], [-2, -2, 3, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, -2, 3, 3, [1], -2, -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [-2, 3, -2, 3, 3, [1], -2, -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, 3, 3], between , [[], [-2, -2, 3, -2, -2, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [-2, 3, 3, -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [-2, 3, 3, -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Here are the surviving meta-words after the first pruning {[3, [1]]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[3, [1]]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, 3], [3, 3, -2], [3, -2, [1], 3, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, 3], [3, 3, -2], [3, -2, [1], 3, [1], -2]} Let's examine them one by one Let's consider , [3, -2, 3] . and enclose it inside , [[], [-2, -2, 3, -2, -2, 3, -2]], getting the word [3, -2, 3, -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [3, -2, 3, -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 3, to position , 8, is a mishap, being , [3, -2, -2, 3, -2, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, 3, -2] . and enclose it inside , [[], [-2, -2, 3, -2, -2, 3, -2]], getting the word [3, 3, -2, -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [3, 3, -2, -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 2, to position , 7, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this word DOES contain mishaps, we kick-it out. Let's consider , [3, -2, [1], 3, [1], -2] . and enclose it inside , [[], [-2, -2, 3, -2, -2, 3, -2]], getting the word [3, -2, [1], 3, [1], -2, -2, -2, 3, -2, -2, 3, -2] Regarding the meta-word , [3, -2, [1], 3, [1], -2, -2, -2, 3, -2, -2, 3, -2], we have that the factor from position, 4, to position , 8, is a mishap, being , [3, [1], -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 79 Lemma Number, 80 The vertex, [[], [3, -2, 3, -2, -2, 3, -2]], is a clone of vertex, [[], [3, -2]] Proof: The potential mishaps are {[[1, 3], [[2], 3, [-6]]], [[1, 5], [[3], 3, [-7]]], [[1, 6], [[1], 3, [-5]]], [[1, 8], [[2], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 2, -th mishap whose potential factorization is , [[3], 3, [-7]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 3 which according to , CrucialLemma[3], are {[3], [-2, [1], 3, [1]], [[1], -2, 3, [1]], [-2, 3, -2, 3, [1]], [-2, 3, 3, -2, [1]], [-2, 3, 3, [1], -2]} In the second crack, we have to insert all the good words that sum up to , -7 which according to , CrucialLemma[-7], are {[-2, -2, -2, -2, [1]], [-2, -2, -2, [1], -2], [-2, -2, [1], -2, -2], [-2, [1], -2, -2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1], -2, -2, [1]], [[1], -2, -2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 3, -th mishap whose potential factorization is , [[1], 3, [-5]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 1 which according to , CrucialLemma[1], are {[[1]]} In the second crack, we have to insert all the good words that sum up to , -5 which according to , CrucialLemma[-5], are {[-2, -2, -2, [1]], [-2, -2, [1], -2], [-2, [1], -2, -2], [[1], -2, -2, -2], [[1], -2, -2, [1], -2, -2, [1]]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap examining the , 4, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap The potential mishaps are {[[1, 3], [[2], 3, [-6]]]} examining the , 1, -th mishap whose potential factorization is , [[2], 3, [-6]] It has three parts The List of Fixed Words is, [[], [3], [3, -2, 3, -2, -2, 3, -2]] In the first crack, we have to insert all the good words that sum up to , 2 which according to , CrucialLemma[2], are {[3, -2, [1]], [3, [1], -2], [-2, [1], 3], [[1], -2, 3], [-2, 3, [1]]} In the second crack, we have to insert all the good words that sum up to , -6 which according to , CrucialLemma[-6], are {[-2, -2, [1], -2, -2, [1]], [-2, [1], -2, -2, -2, [1]], [-2, [1], -2, -2, [1], -2], [[1], -2, -2, -2, -2, [1]], [[1], -2, -2, -2, [1], -2], [[1], -2, -2, [1], -2, -2], [-2, -2, -2]} after , 3, purges there are the following left-over {} Since this is empty, there are no minimal counter-examples coming from this \ potential mishap This concludes the proof of Lemma Number , 80 Lemma Number, 81 The vertex, [[3, -2, -2, -2, 3], [3, 3, -2]], is empty Proof: We have to prove that all the words in the alphabet {3,-2}, that sum to 0, o\ f the form [3, -2, -2, -2, 3, w, 3, 3, -2] are bad (i.e. contain mishaps) Since the whole must sum to 0, and the sum of the letters of, [3, -2, -2, -2, 3], [3, 3, -2], equals , 4 The word w must have sum , -4 W.l.o.g. we can restrict attention to words w that are mishap-free and that \ do not contain zero-factors By , CrucialLemma[-4], each such word must have one of the following conceivable factorizations where [1] denots a word that sums up to 1 {[-2, -2], [-2, [1], -2, -2, [1]], [[1], -2, -2, -2, [1]], [[1], -2, -2, [1], -2]} Let's examine them one at a time Enclosing , [-2, -2], between , [[3, -2, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, -2, 3, 3, -2] Regarding the meta-word , [3, -2, -2, -2, 3, -2, -2, 3, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [-2, [1], -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, -2, [1], -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, -2, -2, -2, 3, -2, [1], -2, -2, [1], 3, 3, -2], we have that the factor from position, 1, to position , 6, is a mishap, being , [3, -2, -2, -2, 3, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, -2, [1]], between , [[3, -2, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, -2, [1], 3, 3, -2] Regarding the meta-word , [3, -2, -2, -2, 3, [1], -2, -2, -2, [1], 3, 3, -2], we have that the factor from position, 5, to position , 9, is a mishap, being , [3, [1], -2, -2, -2] Since this meta-word DOES conatain a mishap, it gets kicked-out at the fir\ st pruning Enclosing , [[1], -2, -2, [1], -2], between , [[3, -2, -2, -2, 3], [3, 3, -2]], we get the meta-word (or ,if there are no [1]'s , plain word) [3, -2, -2, -2, 3, [1], -2, -2, [1], -2, 3, 3, -2] Since this meta-word does not conatain mishaps, it survives the first prun\ ing Here are the surviving meta-words after the first pruning {[[1], -2, -2, [1], -2]} Let's perform the , 2, -th purge We now have the following meta-words to contend with {[[1], -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, [1], -2], [-2, 3, -2, -2, [1], -2], [-2, [1], 3, [1], -2, -2, -2, [1], -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, [1], -2]} Let's examine them one by one Let's consider , [3, -2, -2, -2, [1], -2] . and enclose it inside , [[3, -2, -2, -2, 3], [3, 3, -2]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, [1], -2, 3, 3, -2] Since this meta-word does NOT contain mishaps, we keep it The survivors after this purge are {[3, -2, -2, -2, [1], -2]} Let's perform the , 3, -th purge We now have the following meta-words to contend with {[3, -2, -2, -2, [1], -2]} Let's fatten-up each of these meta-words , by replacing the left-most [1] by {[3, -2], [-2, 3], [-2, [1], 3, [1], -2]} We get the following set of meta-words {[3, -2, -2, -2, -2, [1], 3, [1], -2, -2], [3, -2, -2, -2, -2, 3, -2], [3, -2, -2, -2, 3, -2, -2]} Let's clean them up by discarding all those that contain zero-factors and mi\ shaps After cleaning up, discarding all meta-words that contain mishaps and zero-f\ actors, we still have the set {[3, -2, -2, -2, -2, 3, -2]} Let's examine them one by one Let's consider , [3, -2, -2, -2, -2, 3, -2] . and enclose it inside , [[3, -2, -2, -2, 3], [3, 3, -2]], getting the word [3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, -2, 3, 3, -2] Regarding the meta-word , [3, -2, -2, -2, 3, 3, -2, -2, -2, -2, 3, -2, 3, 3, -2], we have that the factor from position, 5, to position , 10, is a mishap, being , [3, 3, -2, -2, -2, -2] Since this word DOES contain mishaps, we kick-it out. The survivors after this purge are {} This concludes the proof of Lemma Number , 81 This grammar is , true 1 The weight-enumerator for Loehr-Warrington words is, - ---------- 5 -1 + 10 x The whole thing took, 56.322, second.