Ternary scale theorems: Difference between revisions

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* A.3.ii. Scooting the slice ''I'' to the right yields {{nowrap|''I''''R'' :{{=}} ''T''[''p'' + 1 : ''p'' + 1 + ''k'']}}, also perfect. Since its abelianization is a perfect generator, ''I''''R'' has ''i''-many '''X''''s and j-many '''W''''s.

* A.3.iii. Since ''I''''R'' gains a '''W''' and loses an '''X''' relative to ''I'', the lost letter '''X''' is at the leftmost position of I's window, which is ''p''.

* A.3.iv. Conclusion: ''T''[''p''], the leftmost letter of {{nowrap|''I'' {{=}} ''T[''p'' : ''p'' + ''k''],}} is '''X'''.

* A.3.iv. Conclusion: ''T''[''p''], the leftmost letter of {{nowrap|''I'' {{=}} ''T''[''p'' : ''p'' + ''k''],}} is '''X'''.

* B.1. Now we go back to the original necklace ''w''. Lift each perfect generator window (we have {{nowrap|''n'' − 1}} perfect windows) of ''T'' to ''w''.

* B.2. By the hypothesis that ''w'' has an AS, and since the AS descends to stacking a single generator in the template MOS ''T'', the lifted generators ''g''1 and ''g''2 alternate in their counts of '''Y''' and also alternate in their counts of '''Z'''.

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 * A.3.ii. Scooting the slice ''I'' to the right yields {{nowrap|''I''<sub>''R''</sub> :{{=}} ''T''[''p'' + 1 : ''p'' + 1 + ''k'']}}, also perfect. Since its abelianization is a perfect generator, ''I''<sub>''R''</sub> has ''i''-many '''X''''s and j-many '''W''''s.
 * A.3.iii. Since ''I''<sub>''R''</sub> gains a '''W''' and loses an '''X''' relative to ''I'', the lost letter '''X''' is at the leftmost position of <i>I</i>'s window, which is ''p''.
-* A.3.iv. Conclusion: ''T''[''p''], the leftmost letter of {{nowrap|''I'' {{=}} ''T[''p'' : ''p'' + ''k''],}} is '''X'''.
+* A.3.iv. Conclusion: ''T''[''p''], the leftmost letter of {{nowrap|''I'' {{=}} ''T''[''p'' : ''p'' + ''k''],}} is '''X'''.
 * B.1. Now we go back to the original necklace ''w''. Lift each perfect generator window (we have {{nowrap|''n'' &minus; 1}} perfect windows) of ''T'' to ''w''.
 * B.2. By the hypothesis that ''w'' has an AS, and since the AS descends to stacking a single generator in the template MOS ''T'', the lifted generators ''g''<sub>1</sub> and ''g''<sub>2</sub> alternate in their counts of '''Y''' and also alternate in their counts of '''Z'''.