Ternary scale theorems: Difference between revisions

Line 103:

* C.1. Importantly, deleting '''X''''s gives windows of length ''j'', such that when you project adjacent lifted generators (by deleting '''X''''s) to the binary necklace {{nowrap|''U'' :{{=}} ''E''<sub>'''X'''</sub>(''s'')('''Y''', '''Z''')}}, the resulting ''j''-step windows in ''U'' are adjacent and do not overlap.

* C.2. Moreover, for every ''j''-step window {{nowrap|''U''[''q'' : ''q'' + ''j'']}}, there exists an {{nowrap|(''i'' + ''j'')-step}} window {{nowrap|''s''[''r'' : ''r'' + ''i'' + ''j'']}} such that {{nowrap|''s''[''r'']}} is the non-'''X''' that corresponds to {{nowrap|''U''[''q'']}} under step deletion. Since by subclaim A, the unique imperfect {{nowrap|(''i'' + ''j'')-step}} window in ''s'' begins in an '''X''', we know that {{nowrap|''s''[''r'' : ''r'' + ''i'' + ''j'']}} is perfect.

* C.3. We need only stack {{nowrap|2''b'' ≤ ''n'' − 1}} generators (to get {{nowrap|2''b''-many}} ''j''-step windows downstairs) to witness the alternation. Under the ordering induced by this stacking, the 1st ''j''-step subword of ''U'' and the {{nowrap|2''b''-th}} ''j''-step window differ due to parity. Since {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}, this visits every note of ''U''.

* C.3. We need only stack {{nowrap|2''b'' ≤ ''n'' − 1}} generators (to get {{nowrap|2''b''-many}} ''j''-step windows downstairs) to witness the alternation. Under the ordering induced by this stacking, the 1st ''j''-step subword of ''U'' and the last ({{nowrap|2''b''-th}}) ''j''-step window differ due to parity. Since {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}, this visits every note of ''U''.

'''Claim 2''': If a binary necklace ''U'' has ''b'' '''Y'''s and ''b'' '''Z'''s, {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}, and consecutively stacked ''j''-steps in ''U'' occur in 2 alternating sizes, then {{nowrap|''U'' {{=}} ('''YZ''')<sup>''b''</sup>}}.

@@ Line 103: / Line 103: @@
 * C.1. Importantly, deleting '''X''''s gives windows of length ''j'', such that when you project adjacent lifted generators (by deleting '''X''''s) to the binary necklace {{nowrap|''U'' :{{=}} ''E''<sub>'''X'''</sub>(''s'')('''Y''', '''Z''')}}, the resulting ''j''-step windows in ''U'' are adjacent and do not overlap.
 * C.2. Moreover, for every ''j''-step window {{nowrap|''U''[''q'' : ''q'' + ''j'']}}, there exists an {{nowrap|(''i'' + ''j'')-step}} window {{nowrap|''s''[''r'' : ''r'' + ''i'' + ''j'']}} such that {{nowrap|''s''[''r'']}} is the non-'''X''' that corresponds to {{nowrap|''U''[''q'']}} under step deletion. Since by subclaim A, the unique imperfect {{nowrap|(''i'' + ''j'')-step}} window in ''s'' begins in an '''X''', we know that {{nowrap|''s''[''r'' : ''r'' + ''i'' + ''j'']}} is perfect.
-* C.3. We need only stack {{nowrap|2''b'' &le; ''n'' &minus; 1}} generators (to get {{nowrap|2''b''-many}} ''j''-step windows downstairs) to witness the alternation. Under the ordering induced by this stacking, the 1st ''j''-step subword of ''U'' and the {{nowrap|2''b''-th}} ''j''-step window differ due to parity. Since {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}, this visits every note of ''U''.
+* C.3. We need only stack {{nowrap|2''b'' &le; ''n'' &minus; 1}} generators (to get {{nowrap|2''b''-many}} ''j''-step windows downstairs) to witness the alternation. Under the ordering induced by this stacking, the 1st ''j''-step subword of ''U'' and the last ({{nowrap|2''b''-th}}) ''j''-step window differ due to parity. Since {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}, this visits every note of ''U''.
 '''Claim 2''': If a binary necklace ''U'' has ''b'' '''Y'''s and ''b'' '''Z'''s, {{nowrap|gcd(''j'', 2''b'') {{=}} 1}}, and consecutively stacked ''j''-steps in ''U'' occur in 2 alternating sizes, then {{nowrap|''U'' {{=}} ('''YZ''')<sup>''b''</sup>}}.