Ternary scale theorems: Difference between revisions

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* 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'''.

* B.2. By the hypothesis that ''w'' has an AGS, and since the AGS 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'''.

* B.3. For a MOS binary word, the count of a given letter in a generator is coprime to the total count of that letter in one period of the MOS. By this fact applied to ''T'', {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}.

* B.4. Hence, since every instance of the generator in ''T'' has ''j''-many '''W''' letters, every instance of ''g''1 and every instance of ''g''2 has ''j''-many non-'''X''' letters.

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 * 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'''.
+* B.2. By the hypothesis that ''w'' has an AGS, and since the AGS 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'''.
 * B.3. For a MOS binary word, the count of a given letter in a generator is coprime to the total count of that letter in one period of the MOS. By this fact applied to ''T'', {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}.
 * B.4. Hence, since every instance of the generator in ''T'' has ''j''-many '''W''' letters, every instance of ''g''<sub>1</sub> and every instance of ''g''<sub>2</sub> has ''j''-many non-'''X''' letters.